Re: im 21 and have injuries that prevent me from doing some workouts.
I also broke my right wrist two years ago. I am 56 years old, so I know you can do what I did. I was casted from my finger tips to my bicep for 6 weeks and then in a splint for another month. My break was pretty severe.
As soon as the cast came off, I went for occupational therapy 5 days a week. It was intense and painful, but I finally did regain my range of motion and grip strength. I don't think it's too late for you to try some therapy to regain these yourself. You are awfully young to just give up on your wrist.
As far as push ups, I can now do them fine on my wrist, but when I was rehabbing, I used push up handle bars to keep my wrist straight. I still use them on days when I'm doing a lot of push ups at once.
How is your grip? Can you hold a dumbbell? I couldn't hold one for about 4 months, but slowly worked back up to where I am now able to curl a dumbbell and a barbell as heavy as before (For me that's 40 lbs.) I think you're going to have to experience some pain with therapy before you make any gains.