Unless things have changed since I was an undergrad, the scholarships are only full tuition. They are not “full rides”.
My work-study job was for Northwestern University’s admissions office for the coordinator of financial aid for entering students. (I.e. Freshman and Transfers) We had scholarship athletes still apply for financial aid to cover room and board, travel, books, supplies, etc.
Granted, they were few and far between, since tuition was the large expense, but I’m guessing other Div 1 schools had similar financial aid office policies.