Honestly I think you have a choice to make. Let your anxiety hold you back from putting yourself out there and accept that it means your social life will be limited or take the needed initiative. Of course you can talk to a counselor to get support in your efforts and to help you devise strategies to work on this.
I also think a great way to belong somewhere and meet people/make friends is to get a job. Working together forces relationships and the employees all “belong” at that establishment.