I just wanted it to update my case a little bit for anybody who's also going through this.
One month after my injury, I went to see a PT and after some test, he didn't think it's a tear but more like a pinch. I chose to trust him more because 1. I'm going to do PT anyways even it's a SLAP tear, 2. the doctor I went to see before didn't even do any exam on my shoulder but sent me directly to the MRI and when I asked him to write a prescription for my PT(it's required in Texas), he wrote tendinitis.
I have been doing PT for 6 weeks, twice/week and has three more visits left. I made some progress on my shoulder. I already started to climb and I can climb most of the routes as before my injury (I'm not a strong climber anyways). Only have some sharp pain when doing dead hang or overhead reachy moves.
Having trouble doing overhead reachy moves is a very good indicator of a shoulder labral tear. Funny, how many docs send you off to get an MRI before they even give you a full range of motion studies. That's a good way of finding out what's actually wrong. MRI = Day Dreaming. Pseudo-science. Black Art.
This is a late reply. But I am an Class V Whitewater kayaker and I tore my labrum the same. I had the option of surgery or not. I underwent surgery and its the best thing I ever did. It hurt like hell and took like 8-9 months to heal, and I went crazy wanting to get back to paddling but it was well worth it. I now climb with no pain or issues and it feels stronger than my other one. My injury was enough to where sleeping at night and putting on coats etc hurt, so I justified it. I highly recommend it if it causes you problems in the future. Do NOT go to a shitty doctor though. Cheers.