Nah. I'm not sure if your chain will fit through the pulleys on a 9 or 10 speed derailer, but those will probably be priced so high that you'll look at other options instead. Any 5-6-7-8 speed derailer will work fine.

Long cage will work. Medium cage will work. Short cage should work, but I won't guarantee it. The Schwinn site I linked before says you have a 14-28 freewheel. I'm currently running a short cage derailer with a 28 tooth top gear. I have no problems, but I'm not sure how much more it could handle.

One last thing. Are you sure your derailers are actually broken? Or are they just not adjusted properly? The front one looks fine to me, but I know pictures don't tell the whole story.