Pam, you might try a trick I've used to stiffen the bag enough to prevent sagging. Get your hands on a sheet of corrugated plastic - the same stuff that is used for political campaign yard signs (they should soon be readily available for no cost ; )
Cut the plastic sheet to fit the interior width of the bag and long enough to cover the bottom and extend up onto the front wall of the bag. The corrugations need to run across the width of the cut sheet so you can fold it to curve up the front wall. The sheet should be sized to fit slightly snug to prevent it shifting around inside the bag. It's a low cost, light weight solution that repurposes bit of plastic that is otherwise destined for the land fill.
Your Schwinn looks like a sweet little bike in its own right.
Steve in Asheville