Only waiting a month or so for TMS ain't bad. Time crawls when it comes to getting access to mental health care unless you are rich. I hope it works for you.
I have been on a colossal number of different drugs and the only one that ever improved my social skills was Adderall XR. It cleared out the fog in my mind, made it easy to find the words to say, and they just flowed right out of me. I would say it fixed a lot of my social deficits. I watched many hours of sitcoms to assimilate speech/jokes etc from. I practiced my social skills mostly online where I'd meet someone online first, establish a connection before meeting them in person. I had several relationships while I was on that med. Unfortunately I couldn't tolerate the side effects of loss of appetite and altered taste so I quit it. I was on it for ADHD though not specifically for social skills, it just so happened to help in that area that is all though Adderall is used occasionally to treat depression in refractory cases like yours. None of the effects were permanent and my head has returned to a garbled mess, I'm back to being socially awkward, though maybe not quite as bad as where I started.
Maybe consider learning CBT exposure therapy, find some groups or meetups with shared interests that you can attend and practice your social skills there. Unless you are online on dating websites or meeting new people on forums or Facebook, you aren't going to meet anyone sitting at home. You have to be out and about in order to run into people that might potentially be a match. I found if you start from a group of people with shared mental illnesses, they tend to be less judgmental about that seeing as they have one themselves, and you might get along better than with a normal person without any issues.