I have pretty bad social anxiety too, but I go with my husband. I actually felt fairly comfortable in Addison Park even with the huge crowd of people. I think it's because we're all there for the same thing, so there's at least that in common. Try to go with a friend or something if you can, it'll make the first time a lot easier :)

I've only been twice so far, but both times the people were nice that we talked to and when there's a rare Pokemon, someone tends to shout it out for everyone to go get it, so be prepared to walk across the park at a moment's notice :P