Don't eat in the museums. Way expensive.
Take a walk to the Old Post Office. There's a food court in there with just about any kind of food you want. I love the samoas from the Indian place.
There's also places like PotBelly for soup and sandwiches, the usual fast food joint, and if you want to dress up and do a really fancy (and a bit $$) dinner, you could go to Ruth's Chris Steakhouse.
If you like seafood, try the buffet at Philip's Seafood.
If you like Thai, consider taking the Red Line out to Bethesda for Tara Thai or come to Wheaton for Nava Thai. There is also a good Thai place in downtown Silver Spring as well as a fun place called Eggspectations. There might be other places closer to DC, but I don't usually eat dinner in the city.
P.S. My stepson has a high gag reflex, too, and as an adult we discovered it might be fixable with minor surgery, which he hasn't gotten yet. But I understand the limited menu thing.