If you’re thinking about pre-med I would take a very close look at Xavier in New Orleans, LA. They are well-known for producing the largest number of black med school students and doctors and doing it with students who would not have originally been accepted to undergrad at a school like Harvard. They don’t meet full need, but I would run the Net Price Calculator and see what things look like.
I would also take a look at this excellent post about HBCU institutional scholarships: HBCU Institutional Scholarships for Students, which includes many potential full rides. Depending on what your standardized test scores look like, you can let us know if any of these scholarships look like possibilities. If you’re interested in applying to more than one HBCU, you may want to look into the Common Black College App where you can apply to any number of 67 HBCUs for $20.
In addition to UVA and Va Tech, you will probably want to add some other in-state publics. Fortunately for you, Virginia has many good public options. Virginia State (3900 undergrads) and Norfolk State (5k undergrads) are both public HBCUs that are likely to provide a more intimate environment. George Mason (27k undergrads) is known for international relations, and its location a heartbeat away from the nation’s capital means there are many options for political and policy involvement. Virginia Commonwealth (21k undergrads) is less than a mile from the Virginia State Capitol which provides great access for any policy/public health internships and experiences you may want to pursue. James Madison (20k undergrads) also has a lot of students in your majors of interest.
Washington & Lee would be a reach, but they’re trying to diversify their campus and they meet the full need of students (as it defines need).
If you let us know more of what you’re interested in a college such as size (big, medium, small), location (parts of the U.S.), type of location (urban/suburban/rural), the kind of atmosphere you might prefer (pre-professional, intellectually curious, big sports enthusiasm, importance of Greek life, etc), and any other considerations you have, that would help in providing suggestions to you.