While I'm not a tax professional at all, I can relate how my situation concluded.
I ended up getting a corrected W2 (c) form from my company, which removed the NJ tax. It took a while and was a big pain to get processed (corporate bureaucracy).
I think one of the factors that swayed things in my favor is that, due to the pandemic, I was never in the NJ office for tax year 2021 AND 2022 (just NY, work at home). I let 2021 slide and just paid the tax, but once they moved my official office in Dec 2021 I decided to fight it for this year's return.
Not sure how much that matches with your situation or if there's something special about VA/DC tax scenario, but I think a lot hinges on where work is performed and how much you want to fight it (since you probably will need a corrected W2 to get the money back from one of the states/territories). Good luck regardless!
