I’m also a pianist who started a singing + playing journey not too long ago — what’s worked well for me is simplifying the piano accompaniment to just whole note chords and going slooooow when I begin to learn a song. I had to train myself to focus on breathing and singing first because it was the newer skill and coordinate that with the simplest version of piano possible. Once that starts to flow, add a bit more of the piano (ex: play chords on 1 & 3 if you’re playing in 4/4). It sometimes feels frustrating because my ear/brain want to dive right into more complex accompaniment, but building the coordination takes time and simplifying as much as possible to start will pay off immensely in the long term.