Go back to the doctor you saw before and ask to be referred to Femi Ayeni at HHSC. Don't cancel the first referral, tell him you can't take the wait and want another referral so that you can take what comes up first.
Femi is the team doctor for the TiCats and has published extensively on shoulder procedures. He is site chief for orthopaedics and teaches arthoscopy. The other choice is Matt Denkers. Also site chief of orthopaedics (a different site). Bios below:
Bio (mcmaster.ca)
Bio (mcmaster.ca)
As soon as you know that the referral has been sent, call the assistant (Penny for Femi, Kelly for Matt) and be exceedingly nice to them (both are sweethearts).