I've got two probably pointless tidbits of info pertaining to your situation:
Firstly, I've definitely been in your shoes! Haircuts with talkative hairdressers can be so exhausting. A few years ago, I kind of had an epiphany and realized that of all people I might socially interact with, I care the least about how my hair-cutter perceives me. Once I convinced myself of that fact (the hardest part, no doubt), I actually began treating each haircut as a no-risk situation where I could practice my social skills. I went into it with zero expectations and did my best, which left me feeling more and more confident each time.
Secondly, and this is, of course, hotly contested by some...an anti-depressant (Paxil, for me), made an absolutely enormous difference for my social anxiety and GAD. It took at least a full year of treatment before I truly felt a difference, but now, a few years in, and I realized that getting on Paxil was the greatest decision of my life. Things like haircuts, doctor and dentist appointments and so much more that used to literally ruin the preceding week with stress, now feel manageable. There are drawbacks, but you've got to weigh the pros and cons personally. Best of luck!