Am I overreacting to my coworker???

The thing is my cousin works there too and she is his friend already and knows him a lot better but they don't really seem to meet up? He's not making me feel weird at all it's just the situation that I feel very nervous and unsure about.

If he asks you again to meet up, invite someone else along (like your cousin and maybe other people from work). That will keep it platonic and make it less awkward / less pressure for you. I would also keep it to coffee and maybe pub after work, again with others from work, but nothing more than that unless you become friends. If he asks for a more date-like than work-like thing just tell him that you have social anxiety and that you don't like going out that much. If he insists, just insist back in a friendly way.

Be friendly - it is good practice. :-)

Thank you. The thing is I've kind of decided that hanging out outside of work isn't something I'm currently interested in and I don't want to perhaps sour the situation by telling other people - even my cousin, since it might put her in a difficult position as my relative and his friend - so I'm just going to decline that.

It's just the constant Facebook conversations I'm not comfortable with too, since I have found that interactions on there with people you're not already good friends with make things difficult and I need some space as well. Like I don't even message my close friends every single day. Do you think just not responding but still being friendly in person gives the right message?

Not responding at all may be a bit much, not sure. I'd only do that if you dislike him or feel really uncomfortable. But he doesn't know how much you are online or how busy you are, so if necessary just pretend that you don't check facebook every day. You can wait a few days and then drop it into a fb conversation like "gosh only just saw your message, so sorry, don't check here that often / am pretty busy these days. how are you doing etc," and then it's fine if you only reply every three or four days or so because he'll just assume you are busy. Then he'll probably lose interest. It is a bit contrived mind, but it should work / slow him down. Only do it if you can't be found out though, in that case just go with the truth.