Emory might be another school you want to try for. It’s very competitive for admission, so being awarded a Woodruff Scholarship is even moreso. Emory’s in Atlanta (big city) and is part of a consortium whereby students can classes at other Atlanta schools, like SCAD (Savannah College of Art & Design), Georgia Tech, Spelman, etc. And it offers an East Asian Studies major that has a concentration on Korea.
U. of Southern California offers an East Asian Studies major and East Asian Languages & Cultures major, and focusing or Korea/Korean is possible in either one. USC offers half-tuition scholarships for National Merit Finalists, and though I think it is possible to have scholarships stack, I don’t see anything that goes beyond tuition, so you would still be on the hook for room & board.
Also, if you’d consider Wellesley, I’d also consider Harvard. Have your family run the Net Price Calculator (NPC) there. Harvard’s about the most generous school in defining need, so there might be a pleasant surprise. If there’s not, though, no other need-based school is likely going to work.
Indiana University is not in a big city, but it’s in a surprisingly cosmopolitan one (lots of ethnic restaurants including some Tibetan ones, Broadway touring productions come to the campus, it has a world-class music school, etc). Its East Asian Languages & Cultures major allows a focus on Korean, and it also has a competitive full-ride program, the Wells Scholars. More information on the Wells Scholar nomination process is available.