I'm not really like anything I try with my right hand. It all just sounds dull. Even if I walk up and down the chord with my right hand it's a bit boring. So I try playing two notes at a time, and it's a little better, but I think I need some right-handed fingerings for melodies.
Turn the question on its head. What are the chord progressions good for, if they aren't supporting a compelling melodic line? (That's not to say that there isn't a place for chords that exist in their own right, but a piece that is purely chordal throughout its length is a tour de force.)
Chord progressions tend to be either melodic lines superposed on one another in a fashion that creates harmonies, or harmonic support for a leading melody, or some combination of both. (Sometimes melodic figures implied in the accompanying voices of a progression can give you the elements of a very successful melody - in effect, you get a melody and accompaniment with some rudimentary hints of counterpoint).
That being the case, it might be wise for the moment to start from melody. Create melodic phrases that aren't improvised to create a foundation for improvisation, then work out where the consonances and dissonances are implied, and harmonise the phrases accordingly. That's frequently how the standards work: the melody gives the nucleus of the arrangement, and the arrangements not infrequently re-harmonise the tune (as compared to the original). In good music of any sort, melody, even improvised melody, isn't an appliqué or add-on; it's integral.
The advantage of starting this way is that you will absolutely have to treat some of the melody's notes as non-harmonic (that is, as suspensions, anticipations, passing notes, appoggiature, etc.), because changing chords with each change of note in the melody is going to get awfully old awfully fast. Harmony has its own rhythm, and it usually takes a more deliberate pace than the melody.
Then too, elements like scale segments, leaps, dissonance and resolution, syncopation, suspension and so forth all give melody its drive and flavour. Right now, because you are letting chord tones dictate the melody completely, you aren't getting any of that, so work "melody first" until conceiving melodies with those characteristics becomes second nature. Once you have reached that point, the kind of idea you start with, whether chordal or melodic, won't matter, because each will call up for you a version of the other that has some character that you can work with.
Good melody has structure: it has motifs and phrasing, and both repetition and variation. Figure out where your melodic "hooks" are, because they will give you something to come back to. Melodic phrases are like harmonic phrases - they have cadences of varying strength - so you will want to be very careful about not coming to rest on the tonic, or a note of the tonic chord, too hard before you're ready to come to a halt.
Pay close attention to the "roof" and "floor" of your melody, its high notes and low notes: these give the long range direction of a melody, both to and away from climaxes. They not infrequently establish long-range dissonance as well. For instance, if you have established the key of C and, after a time, hit and leave hanging a high B (taking the melody immediately lower for a bar or more), coming back later to a high C, everything in between will tend to act as a suspension of the B, with the high C acting as a resolution, even if the B was sounding as a consonant chord tone, even if the melodic passage in between was harmonised with its own logical set of chords. That sense of dissonance and resolution will tend to bind the entire passage together and give it some urgency.
So, to give a melody a sense of directed movement, you absolutely need to a) set goals for it, and b) create a sense of tension and release to achieve those goals.
The goals can be short-range (such as walking a phrase up from the tonic to the dominant) or long-range (e.g., widening the range of the melody progressively over the course of several phrases to climax on a high note). You need to be aware of both levels.
The tensions can likewise be short-range (for example, a note that is dissonant to the underlying chord and resolves immediately to a chord tone) or long-range (such as my "hanging B" example, where the note is in tension, not with the underlying chord, but with the prevailing key - in the key of C, a prominent B in the melody implies movement to C). Again, you need to be aware of both levels.
EDIT: Here's an example, a somewhat syncopated melody over simple comping in the left hand. I created it on the fly, so I'm not going to claim any great artistic merit for it, but it does illustrate how to keep a melody moving.
The "hanging B" is in blue, and its melodic resolutions are marked by the blue bracket above the stave. The A is left black because it is a minor sort of resolution; the entry of the C in blue is quite noticeable. Notice that these two "resolutions" are approached from below by leap (of a third and fourth respectively) to set off the motifs they start. It's the fact that these notes do follow from B that justifies the leaps.
You'll see a similar situation in the melody's "floor", here marked out with green. The low C is left hanging by a leap, to be picked up by low B in m.3. If you look at the high notes and low notes in the first two-and-a-half bars, they form a wedge leading to the b'-b octave in the melody. The high notes in general move quite steadily from g' at the start of m.1 to the high f'' in m.7.
This is the kind of thing I'm talking about when I talk about roof and floor - the high notes and, to a lesser extent, low notes of a melody stick out and define the melody's shape, so they should have some logic behind them. That high F, by the way, is just begging for another high note to follow it, at least in part because it marks one end of a tritone scale segment (to B in m.8).
Now, if I were continuing this piece, I'd probably leave B (in m.8) hanging again, start up a broken chord figuration in the left hand, bring the right hand melody right down into the top of the bass stave, and elaborate the suggestion of having moved to E minor. I'd also start moving the melody slowly towards using steady quavers (eighths) and semiquavers (sixteenths). The slightly unstable high F and the hanging B would be my "hooks" to haul myself back up into C major again. This is the sort of thing I mean by "long-range goals". Even when improvising, you get a very good idea of where you're going to be 8, 16, even 32 bars down the road.
When composing or creating a melody line to go with your chord progression (harmony) your safest bet is to choose a chord tone for the first note played after or simultaneous with the chord change. A chord tone is any note contained in the underlying chord (one of the three in a triad or one of the four in a 4 note chord such as a 7th).
But you can include non chord tones for notes in your melody to play over a certain chord. Ideally, the non chord tones (notes that are not part of the underlying chord) should consist of notes in the primary key and should resolve quickly to a chord tone or to a note in the next chord in your progression.
When choosing notes for your melody, a good rule of thumb is to use primarily the notes that are in the underlying chord, and limit most of the passing non chord tones to notes that are both within your primary key, and part of the diatonic scale based on the chord you are playing. The chord you are playing as part of the harmony will suggest an emphasis on the chord tones - so they should be used more than the non chord tones while playing a particular chord.
So if we were to compose in the key of C major for example, we would likely start our chord progression with a C major chord. Our chord tones for C major are C,E and G - so any of those notes will work in your melody while playing the C major chord. Our options for notes (used sparingly as leading or passing tones) other than the C, E or G - are any note in the C major scale (key of C major). This works because C is the tonic chord (the chord that is the same as the key we are playing in).
Another chord we would probably use in our chord progression in C major would be an F major. The notes for an F major chord are F, A and C. The same guideline applies to make most of the notes used while playing the F major chord - either an F, A or C (any octave).
The other notes available will be derived from a combination of the F major scale and the C major scale. We want to choose from notes that are common to both scales. We want to keep our notes within the C major scale because our song is in the key of C and if we are singing or playing melody notes over an F major chord, they will sound better if they are in the key of F major (found in the F major diatonic scale).
The notes in C major are C,D,E,F,G,A & B. The notes in F major are F,G,A,Bb,C,D & E. The Bb note is not in the key of C major so if used it would be an "accidental" in the key of C major. The B note is in the key of C but will not harmonize well over an F major chord because it is not in the key of F major. This leaves the notes common to both the C major scale (key of our song) and the F major scale (the chord we are now playing) which are F,G,A,C,D and E. So our available non chord passing tone notes (that will harmonize well with an F major chord) are G, D and E.
One way to add more interest to your melody lines is to create a bigger sense of movement by using octave wise jumps in your melody line. For example - the song Over The Rainbow originally written for The Wizard of Oz uses an octave as the interval between many of the melody notes.
These are just a few ideas. Hopefully other's will add more ideas as well.
This is a bit old-fashioned but still useful. (You can modify the ideas to fit your needs.) https://ia802605.us.archive.org/16/items/exercisesinmelo00goetgoog/exercisesinmelo00goetgoog.pdf To some extent, the general pattern of melodies are just moving up and down scale lines and arpeggios. The book above does give some exercises you can try to see the difference between accenting a chord-tone in the melody or a non-chord tone. The author also gives some good hints of how to make big skips in melodies and why.
There are also good ideas for note lengths.
These things just take practice. Try different ideas. See which ones sound good (or at least sound like you want.) The book gives some good starting places.
There are other books on the subject, but this one is extensive and free. I like to review it now and then just to get some new ideas.
I read a Brahms quote once, he was staying at a hotel that housed a large number of musicians, and he apparently found it stimulating to hear a variety of people practicing and playing music all day long. He wrote in a letter that the most wonderful melodies were just appearing to him from the air.
There is a very nice series of books by Bradley Sowash that teach how to improvise on piano. He goes very step by step.