I am right there with you. I also play guitar and sing and write songs. Lately I have been trying to play more piano but don't want to devote the time to take lessons. If you follow the method detailed below, you will be accompanying your singing on piano in a surprisingly short period of time.
First - remember when you first learned to play guitar? You learned a few chords - perhaps a G and a C and a D chord. Once you learned to play those three chords you could play any song that used only those three chords. You were playing guitar and building your repertoire!
Then one day you found a song that had the three chords you had memorized and mastered but also had an Am (or perhaps it was Em) chord in it. So you learned to play a new chord. Now there were even more songs you could play! Then you learned an F and could play songs in the key of C.
Gradually you learned all the main guitar chords to play pretty much anything you wanted to play. And now if someone says play a G chord or a G7 or Cm or whatever - you don't hesitate, you just play the chord because you committed the fingering to permanent memory.
Guess what? You can play all the same chords on piano that you play on guitar. And guess what else? You don't have to make your fingers and wrist and hand curl into strange contortions to play them.
Go about learning piano chords the same way you learned guitar chords, one at a time. You will find the learning curve is easier because you don't have to master the contorted fingerings so much as just memorizing the shapes and positions. Practice the common chord changes (like on guitar) until they are smooth and you can do them without hesitating.
Start by learning all the chords for a simple three chord song you want to play. Then add more. Learn all the basic chords in your favorite key to sing in. Then learn the basic chords in your second favorite key.
Learn to play the right hand chords together with the root note of the chord with the left hand (same way you play guitar by using both right and left hand). You won't always play them simultaneously - but learn to position your left hand over the root note so you can mix the left and right hands together as the rhythm dictates.
I like to play the root note with my left thumb and an octave lower root note with my left pinkie and sometimes alternate back and forth in a rhythmic fashion. This is something you can add after you master just playing the root note with one finger. It's a little more challenging because you have to move your hand around more - but it will mostly be moving in parallel to your right hand (same direction).
Every now and then I get fancy and throw in another chord note on the left hand with my middle finger. But that will come with time if you just focus on positioning your left hand in concert with whatever chord you are playing with the right hand.
You can download and print free piano chord charts from the internet. Print out the ones you need and put them on your piano's music stand and practice playing them. Apply them to a 3 chord song you want to learn. Once you learn that song, you will have those three chords in memory and you can build from there.
Remember how the first chords you learned on guitar were the easy first position or open chords? On piano, start with the basic first position formations. Once you are playing songs with the basic chord, learn the inversions. There are some really good inversion charts that show the basic chord and then the first and second inversion. Find a song where one of the inversions of a chord you learned seem to sound better and learn to play that song with the inversion to add that new shape to your arsenal.
After you master most of the basic chords you can practice adding embellishments by playing the chord notes individually or throwing in some other filler notes like you do on guitar. This is actually easier than picking individual strings on the guitar while playing a chord. For major triads - try playing the octave of the root and alternate between inversions depending on what sounds good in your song.
Learning chords on the piano will not take as long as on the guitar because many of the hand positions are similar from one chord to another. It's just a matter of placing that shape in a different place on the keyboard, For example - you can use the exact same shape to play a C or an F or a G chord just move the shape around like barre chords on guitar.
I like to call this a chord based method of learning piano to accompany singing. You won't be playing classical pieces by famous composers but you can certainly accomplish what you stated in your question. And you won't need to read standard notation or sight-read polyphony. All you need to know are the chords - just like on guitar. And you will learn to improvise from there.
I have included some pictures below that I found on-line to show some examples of the type chord charts you can expect to find including an example of an inversion chart to use once you master the basic chords. There are plenty of different type charts available. Choose the ones easiest for you to read and relate to.
You will be playing piano and singing along to your favorite songs in no time at all.
EDIT - Please note that the author of the image above chose to use the words "correct fingering" when in fact a more accurate label would have been "possible fingering" or "suggested fingering". Some of the numbers that indicate which finger to use to play the indicated notes might represent a typographical error.
Use whatever fingering makes the most sense and is most comfortable for you and don't pay attention to the fingering suggestions in this picture! Also it is important to know that the red colored key is not part of the chord but is just showing where middle C is located for reference.
The image below is an example of the type of info you might find online to show how to play inversions. The example below is for a C Major chord.