I live in Bridgeyard on the south side of Old Town. It's within your price range and the walls are the thickest ever seen in an apartment building. Like I literally can't hang pictures with a nail because they're made of cement or something. I never hear my neighbors at all. The apartments themselves are pretty Spartan (no in unit laundry) and the building has a roach problem, but the location is amazing. It's right next to Jone's Point park and the Mt Vernon Trail and there's the first stop of the 10B bus line right next to it that will take you Braddock Rd station in about 10 minutes. Leaves every 30 minutes. You can also walk to King St Station in about 25 minutes if you don't mind hoofin it.