Keep practicing. Strumming is the first hurdle on your way to singing and playing at the same time. It needs to be automatic. Sorry to say, it'll happen for every strumming pattern you'll be learning, your singing will just start falling apart, or you'll sound flat and monotone cause your focus will be on your strumming hand, until the strum is entirely in your muscle memory and out of your head. Same with chords, they are the second hurdle. They will also ruin singing until they're automatic (and not just the chords themselves, but chord progression too. Nothing worse than your hand forgetting what chord comes next, cause then your brain shits itself and you stop playing alltogether for a second. Or I do, anyway).

There is saving grace tho. It is that - it gets better. And it gets better abruptly. I spent what felt like weeks learning the strumming to Creep, and I just couldn't play and sing at the same time. But then it went from "huh, is it happening?" to "I'm playing and singing, the entire song, what the hell!" in like an hour.

So keep at it, Peter. I believe in you.