So, I'm probably not the best example because, despite being dead scared of talking to strangers and social situations, I'm apparently charismatic :P However, I will tell you a few of my past bf/gf's.

One I met at a support group. He asked me out on a date because he thought "I had something to say" and looked smart.

One we were in class together, though we knew each other outside class too. I asked her for coffee after each class, even though it was late.

2 more were in other classes of mine, 1 high school, 1 first year. Although the girl from first year was another old friend of mine. We got together after she helped me get over swine flue!

And one I'm actually NOT interested in, but am fairly sure is interested in me, I met at a friends party (we hid in a corner since we both don't like parties) and he's going to teach me to do jewelry making. The only reason I don't want to date him is personal reasons (getting past mental blocks involving relationships) and that I am leaving soon.

Basically what I'm saying is that your best chance for finding a meaningful relationship is by not really looking, and then going for whatever chances occur. Try some of the activities the others are suggesting. Something therapeutic that you can do for your anxiety just as much as meeting people. Don't try and cover up who you are, but try to just talk to people and get to know them as friends, not objects to be captured. Even if you don't walk out with a gf on your arm, you will have achieved a lot, and will have found people to talk to.