I am working on this exact thing at the moment! I have an exercise that trains your ear to hear when you're successfully singing chord tones. Just before I explain that one, allow me to give you a prerequisite exercise. (And full disclosure, I am only singing a year myself, and struggling with this very topic; this is what I am doing to improve)
Sing a major scale in solfa, but on your instrument give yourself the root of the scale. So play a do and sing a do. Play a do and sing a re. Play a do and sing a mi, etc. Do it fairly slowly (sing each syllable as a whole note in 4/4 at a bpm of quarter note = 80).
Because you constantly have the tonic in your ear from the instrument, you start to detect a very obvious consonance at each interval between what you sing and the tonic. Similar to how you can hear clearly when you sing perfect unison with a note being played, you'll be able to hear when you're singing a major 3rd, or perfect 5th or whatever above the note that's being played. This will help you to tune yourself with your ear, instead of apps.
Then just extend exercise 1 to singing over sustained chords. So play a sustained triad, sing the chord tones, and internalise how each note you sing sounds relative to the chord, exactly like in the previous exercise. But make sure that you also sing chord tones in octaves that are not being played.
Example: If you play a C Major voiced like so: C2 G3 E4, sing over it with the following notes(depending on your range): G2 then C3 then E3 then G3 then C4 then E4. Spread the chord you play wide, narrow, voice the chord so it's played outside your vocal range, etc; whatever helps. You can start by scooping in between the notes you sing until you land on the one you want, or you can stick to a metronome and be more precise. Also you can obviously modulate keys, which goes back to your question.
Finally, one last thing, sometimes the note in the vocal melody will not be a chord tone. Perhaps there's a D MINOR triad (D F A) and you must sing a G. This can get tricky too if the voicing is D3 F3 A3 and you must sing a G3. Best exercise to deal with this situation is to sing major scales over the chords, in exactly the same way as exercise one.