Yes, GW offers full-rides. The only ones I know of for sure, though, are for local-area high school students in low-income families. There could be others offered to students outside the DMV area as well.
As far as acceptance into the BS/MD program lowering your shot at financial aid, I’d guess no, because I’ve heard that aid is independent of which school or program your accepted into. However, if you want to know for sure, you can always talk to the financial aid folks at 202-994-6620. They’re quite helpful and should be able to answer your questions much more definitively.