My recommendation is to check out
Washington Area Scuba Club: A DC/VA/MD Scuba Meetup Group (Washington, DC). The group there plans local as well as out of the area trips. They've got a trip to NC planned for late may.
They're not what you're asking for, but below are some local "I just want to go for a dive" options.
Millbrook Quarry:
Millbrook Quarry (aka Lake Millbrook) - Haymarket VA | DiveBuddy.com
For Millbrook...the visibility is fairly bad. That and from what I've read, it never really warms up during the summer (unlike Lake Phoenix). Millbrook is deep though, unlike pretty much everywhere else in the area.
Lake Phoenix:
Lake Phoenix (previously Lake Rawlings) - Rawlings VA | DiveBuddy.com
I enjoyed Lake Phoenix (I did my OW dives there). It's a nice setup and the visibility is good, but distance wise...the juice just isn't worth the squeeze for a day trip from DC (to me, at least). Once it warms up and I can run down there after work on a Friday, camp out, and spend the weekend...I'll head back down there.
Rappahannock Quarry:
Rappahannock Quarry - Fredericksburg VA | DiveBuddy.com
There's sporadic access to Rappahannock Quarry (Fredericksburg). You would need to contact Scuba Shack in Fredericksburg and see what the schedule is.
I managed to get into the Rappahannock Quarry on New Years day when Scuba Shack held a fun dive. The visibility was pretty good (20'+), their DM/guide was a good guy, and I had a great time. I would definitely go back if I could get in there again.
I went out with Kent Island Scuba a couple times this past November. I had a good time, but I didn't see much...as the visibility was poor. Dives are out on the bay.
https://www.kentislandscuba.com/
On the WAY out of town options...I made the drive down to Florida the middle of January. It was quite a journey. It was ~15 hours down to Lauderdale By The Sea. I then went down to Key Largo (an hour and change), Pompano Beach, and back down to Key Largo.
I intended to head home after Pompano Beach, but I had such a good time that I wasn't ready to go back home just yet.
I would definitely do that trip again, but it was quite a haul down there...so I don't see me making that trek more than once a year.