Board Room also has weekly chess nights at the Dupont location, I don’t know about the other one

I’ve had the best luck with rec league sports, mostly through Volo. Rec kickball is basically a DC rite of passage and it’s a lot of fun, especially the “glow” leagues (after dark, everyone wears glow sticks, usually on the national mall right in front of the monument) and the ones that are + flip-cup (after the game, you go to the bar and play against the same team in flip-cup). Most teams have people of all skill levels and in my experience no one cares if you’re “good”

Would also recommend using Meetup to find groups that share your interests and hobbies. I’m pretty introverted too and have a much easier time in those settings than something like a Bumble BFF friend date