I injured both snowboarding, and I couldn't do pushups with my hands because I couldn't bend them, use your fist and take it easy and you will be fine, if this persist for a long time then you should...
I had the same problem when I started lifting, my wrists got so bad that they were hurting even when I didn't do anything... Then I tried keeping my wrist straight in it's natural position at all...
I thought the same way until I screwed up my shoulder, rehab takes weeks if not months, so I suggest that you do the same amount of pulling and pushing...
Let me explain a little more, I had a similar issue and saw 2 docs, one of them was a super duper sports med doc but they didn't figure out my issue! Until I saw a physical therapist and he was right...
So I hurt my shoulder around 2 weeks ago - doing bench, pull-ups and dips - and I'm going to see the doc on Monday but today I noticed something strange...
So I went to the walk-in clinic and the doc said that my RC is probably inflamed but she said it's not a tear because if it was I would have more pain than what I have and I couldn't raise...
After a lot of research I think it's my RC but the thing is that I can move my shoulder OK, I feel pain in some positions but it's very mild, I can extend my arm to the side and in front with no pain...
So when I was doing bench, my shoulder started hurting me and I stopped, it was painful the last session but these types of pain usually went away for me.