Those Schwinn frames aren't going to be any better than what you have.

Perhaps these frames?
Chrome single speed bike
TORELLI ROAD FRAME 57cm

The chrome frame undoubtedly is a bit cheap, but it may only be a pound or two different from a reasonably high end 80's model frame.

The Torelli frame might be nice. It needs some work to clean it up, but if it was about 2000 miles closer, I might consider it.

I'd encourage you to just use it with the horizontal dropouts, and leave the derailleur hanger and cable stop in place, but it would be a good frame to customize as you see fit.