That is a tough one, I will agree with you. Having a baby to care for is all-consuming (I managed to raise two of them).
And. If you want to write, you’ll find a way. It may not happen at this exact time in your life, but maybe when your baby gets a little bit older it will be easier. The days go by slow, but the years go by fast. For real.
A couple things that might help. Number one, lower your expectations of your writing. Maybe right now you just keep a notebook open and write down a few notes. A sentence here and there. A paragraph! Just something to keep your hand in, so to speak, and your mind engaged.
And second, let those dirty dishes stay in the sink and sit down and write for 10 minutes before you do them. Find a prompt (it can be anything!) and set a timer and go. The dishes will still be there when you are done, but you’ll feel better because you’ve written!
Don’t beat yourself up for only doing a little. Every bit counts. And you are doing the most important work in the world right now — give yourself credit for that.
I hope this is helpful!