I have RSX 7 speed brifters. I had to rebuild the left one and replace a broken spring with one I made myself. THey work great otherwise.
For now, ride the bike as-is. When you have $30 to spare, get a 7 speed freewheel and a new chain, and new cables. It will run like a new bike.
FWIW, every time I see a bike with badly mismatched components for sale, I think "STO-LEN."