I have an 87 Schwinn Prelude in the 19" size. I got it for $125. It has good Shimano components, 700c wheels, Columbus Tenax tubing, braze-ons for racks/fenders, etc. It's a really great bike. Mine weighs about 24 lbs, which isn't much, really.
I wouldn't get a Varsity, personally...there are just so many nicer bikes out there, even if you just want to stick with Schwinn. They come around often enough. it's just that the smallest of them/the nicer bikes, in good condition, usually cost a bit more, at least in my area. At 5'5, you might be able to get a 50-52 or so, depending on your measurements and how you want it to fit, though.
Other brands I would watch for are Miyata, Univega, Trek, Raleigh, Centurion, Nishiki, Specialized, etc...Spend some time over in the C&V sub forum
If you want to see some other Schwinns, though these are from the 80's -
Show off your late 1980's Schwinn road bike here
If you can't see all the pictures because of the photobucket mess, you can download this add-on so you can see em, if you are using Firefox or Chrome.
Firefox -
https://addons.mozilla.org/en-US/fir...dl-upandcoming
Chrome -
https://chrome.google.com/webstore/d...alioapbifiaedg