You'll find most of the major chains in Lake City. But you should make reservations ASAP. We stayed in Lake City on our way down two years ago. It was also a game weekend. Even though we were 90 miles from Tallahassee, most of the hotels were booked solid with fans.
We live 20 miles from Tally, with family in Lake City. I'd book Lake City but fast like PP suggested. Lots of folks will stay that far away for a good game. More to do in Lake City, more food choices etc.