@CRASH is prolly the guy to ask..
We'll see about that.
Ok... a Huret didn't come on that bike. That's probably a big source of your problem. I'll bet someone goobered up the original Shimano and took it to a Schwinn shop.
You have 3 options right now:
1) Deal with the Huret. We'll talk about how.
2) Replace with an appropriate Shimano. We'll talk about which if you need. You said you had one... but do you have the RIGHT one?
3) Break the bike down and send it to me, paying for all shipping expenses.
I personally recommend Option 3.
If Option 2, then tell me what you have, and we'll go from there. (Since I doubt you'll fall for Option 3, this would be my recommendation)
If Option 1: Those smooth guide wheels are a bear. They came on Fastbacks and others, but don't really guide like they should. They're built for a smaller freewheel cluster, but a Spyder has a 14-32 cluster (Some had 14-28) So you're asking a lot from a guide wheel that can only "suggest" where the chain goes. But let's give this a shot.
First, you can't do anything with the bike upside down. Turn it over and either hang the seat from a rafter, or rig up some way to suspend the rear wheel. Maybe a rope? Or... take your 6 years of experience on this forum and go buy a bike stand!! (You'll never know how you lived without one)
Once the bike is set up, you need to eye ball that derailleur. It the cage vertical with the frame of the bike? Is the body twisted or deformed in any way? Is the cage laterally twisted, meaning, is it trying to guide the chain towards the left pedal instead of the sprocket ahead of it? Is the wheel on straight? Are the dropouts bent, or misaligned?
If all that is good, then you can actually start adjusting it. That will be my next post that I'll write after my meeting...