Introducting Laura, one of my students who is a very talented vocalist/pianist, and I'm excited to work with her!
Practice playing first to activate that muscle memory so it almost becomes automatic.
Create a recording of your playing, and then sing to that. Put them together when you are comfortable.
Watch artists on YouTube who accompany themselves on piano, like Nora Jones, Billy Joel or Sara Bareilles, to check out their style of playing while singing.
Focus on vocal production over piano playing. Keep a basic rhythm with chords, instead of playing complicated arpeggios.
Try to keep your head up, especially if using a mic. Position the mic so that you don't feel like you have to stretch your neck.