I've had exactlt the same experiences so rest assured you're not the only one and whether you're socially awkward (I am also and an introvert), these places really can make you feel dreadful. So here's what I decided to do;
I made the decision that I was going to these playgroups for my little girl, not for myself. Honestly your life isn't your own anymore and putting on a brave face for your child really will build up your confidence in these kind of scenarios.
I purposefully moved out of my comfort zone otherwise I'd never get stronger, wiser or make ourselves open to new experiences.
This in turn helps one realise how they want to be treated; if you see a socially awkward mother now's your cue to help her out (perhaps make a new friend?)
You don't have to stay! I tried out a couple of baby groups before I found a few that were somewhat decent. If you're going somewhere where the people don't recognise simple social ettiquette then move on. "Different horses for different courses!" As the saying goes.
Try and find some church-led baby groups (I found a fantastic one). The women who organised it made it their job to talk to every mum there and make even the shyest mum feel welcome.
Do understand that in essence all of us are just trying to survive in a very hostile world and not everyone treats us how we'd like to be treated. I do believe that what you put out, will eventually find it's way back to you so you will find your right people if only for a season or in for the long haul!
Don't give up! And all the best to you and your little one - they grow so fast and this time will pass in a heartbeat.