Many schools on your list will have a DC Semester.
You have U of SC - my daughter attended their DC semester via College of Charleston who uses the U of SC program - but note you have to be in Honors to attend - so getting into U of SC alone isn’t enough.
The brownstone next to theirs was IU (Indiana).
If you want to be in DC or adjacent, it’s great but you don’t have to be. And internships weren’t hard to find - mine had 7 full time offers for her semester, 5 paid.