I don't know of the best veggie options off-hand, but as far as counter service choices:
In DL, I like Bengal BBQ - great chicken skewers. I also like River Belle Terrace for breakfast. Plaza Inn has good chicken. Oh, and if this counts as counter service, I can't get enough of the goodies at Blue Ribbon Bakery. I haven't eaten at Pizza Port in a long time but from what I recall, I liked it.
In DCA, I like Taste Pilots Grill. I have to echo what Pixiewings71 said about the BBQ chicken sandwich. Mine actually came with a ton of BBQ sauce on it so I didn't need to add more (in fact, I wish the chicken breast had been a little bigger), but I did load on the onions. Plus, the fries that the BBQ chicken sandwich comes with are great - they kind of look like a cross between Ruffles potato chips and pickle slices. I was using those to try to sop up some of the BBQ sauce!
In the three hotels, the only counter service-type place I have tried was the little coffee/snacks place in PPH, and actually that was pretty good. Yummy blended mochas.
In DTD, this doesn't really count, but Marceline's has some good sweet treats.
I intend to try Whitewater Snacks and Pacific Wharf Cafe on one of these upcoming visits!