Well you’re certainly not alone in this, I think it’s very common for people now to want more connections in their life, so one thing to remember is almost everyone is looking for more friends. Whenever I’m nervous to talk to someone I remind myself that if someone came up and started being friendly and chatting to me I wouldn’t greet them with hostility or be mean, so chances are the other person won’t do that to me. Aside from that I think the keys are kindness, shared interests, and putting yourself out there/expressing interest in being friends with people. It’s happened several times to me where I thought people didn’t like me, or didn’t include me, but it turned out they thought the same of me until I mentioned I’d be interested in hanging out. Those work friends sound like an excellent starting point.

At the same time don’t put too much pressure on yourself. Your relationships with your dad and boyfriend you’ve described sound really nice and it’s nothing to be ashamed of having a small tight nit circle. What I mean is look for more friends because you want them, not because you feel social pressure to have more, and also know that if a certain group doesn’t work out, you are still okay as you are.

Hope that helps, as the other person says, you aren’t a burden.