The HGI on 14th Street is a fantastic location for sightseeing; you can walk easily to the White House among other spots, and there is a subway station right across the street. Some of the rooms are a little small as the hotel is in one of the older buildings downtown. If you'd like a little more space there is a relatively new (less than 4 years) Embassy Suites on K Street around 10th or 11th NW which might work well for you. It is the same basic neighborhood as the 14th Street HGI, you;d have about a 3-block walk to one subway line and about 6 blocks to another line, so easy to get around from here.
That particular neighborhood is a little spotty after dark so I would avoid using the little park as a cut-through from the subway if staying at the ES. The choices for dining are better in the daytime than the evening, as this is primarily a business district that clears out after 7pm or so; however, you would be very close to several neighborhoods with lots of evening choices, particularly Chinatown (which has about 2 actual Chinese restaurants left). Cabs within the city are cheap; a $6 ride from either hotel would have you in DuPont Circle or the U Street Corridor, a few more $ to get to Georgetown or Adams Morgan.
Keep in mind that parking in the city is quite expensive; I believe the HGI is close to $35 a day if not more. If budget is more important than convenience, some of the suburban Hiltons may be a little more affordable. You'd want one that offers shuttle service to the metro--- Hilton Alexandria Mark Center is one. Hilton Alexandria Old Town does too (I am fairly sure) and is in a really neat historic neighborhood, cobblestone streets, lots of shops and restaurants, but definitely not as convenient as the in-town properties.
Sadly I have not been to Myrtle Beach in ages so I can't help there.