I have a 1980s Schwinn Traveler and was lucky enough to find a 27" aluminum wheelset on a dead Myata. Sounds like my vintage Schwinn was in about the same condition as yours. My fix up included swapping the wheelset ($0), new Bontrager tires and tubes ($45), general cleaning, repack all bearings, a set of Bell cables from WalMart ($10), new brake pads ($15), swapped for the lighter aluminum handlebar (also from the Myata), and new bar tape from a bike co-op ($5), and a used entry level Bontrager seat I had laying around. As I got the bike for free, I have about $75 into a vintage commuter that doesn't look or ride half bad. Not a modern road bike by any stretch of the imagination but well worth what I have into it as a Plan B bike that my stepson rides on a regular basis.
Don't put big money into an old Varsity, just clean it up, repack the bearings and lube the brakes and drivetrain, and replace consumables like tires, brake pads and cables with budget minded parts. You'll be able to ride it for a long time. When you get the funds together for a better road bike, the Varsity will still make a great Plan B / crappy weather / low theft risk commuter. I've owned a few old 10-speeds and they can still be reliable and a lot of fun. I know a guy who last summer did a 600-mile trip on a Schwinn World that was 90% original. In 1976 the big BikeCentennial trek across the United States included many steel 10-speeds so don't let anyone tell you they aren't capable bikes.