I've gone down the list of DC public housing and only the James Apartments at 1425 N St NW, and possibly Claridge Towers at 1221 M St NW (it's on the boundary) are the only public housing apartments in Shaw.
The Northwest Co-Op 1 and 2, not public housing.
Asbury Dwellings on Rhode Island, not public housing.
1330 at 1330 7th St NW (formerly Immaculate Conception Apts), despite the smell of fake weed, not public housing.
The McCollough Terrace Apartments, strangely some of the few non-Suzane Reatig designed UHOP buildings in Shaw, obviously not public housing.
The Washington Apartments, not public housing.
The Gibson Apartments, (maybe formerly public housing?) but not public housing.
Now there is this other thing called Section 8, but that's for another post.