That can get a bit tricky if you are using two-handed signs. I'm sure Hayden and others would appreciate some elaboration on this. Maybe some BSL or Auslan users can pitch in here
Switching over in ASL for the alphabet is simple - since it's one handed (unlike BSL etc)
For the rest - just use your non-broken hand as your dominant and use the broken hand for "recognition assistance" only when it comes to signing (ie keep the neutral handshape, just use the arm for non-dominant "position cues" when need.)
Think how you'd sign if carrying a cup of coffee (ie the broken hand) - it's basically the same thing
im lefty and i broke my hand and wore a cast for 2 months. i signed and write with my right hand for 4 months cuz i broke 3 bones on left hand and had to put 3 steel rods inside of it. it was pretty tough but i went well with my right hand to write and work with.
deafies have their common sense to understand what you talking about with your cast on. i had that experience with no problems to communicate with one or two with a cast on.