I think the main thing is that you need to keep doing the shoulder mobility exercises, but don't allow your wrists to be that extended either way. Take them out to the point where they can be neutral and tighten your back at that point. Your hands will be wider but you won't be taking any of the weight on your arms and your wrists will be much happier.
Aside from that your form isn't too bad. You're getting too far into your knees it looks like (see your heels coming off the ground) and you're pushing these slightly from your toes, so reach your ass back more and shove your knees out to the sides a little harder. Fix those things first.
I personally would still do the program now and the cut later.