Accidentally put my Festiva into reverse without hitting the clutch, now it is making a whistleing noise

Unfortunately it sounds like the trannsmission needs to be repaired or replaced.
You probably damaged a busing on the reverse shaft which is now starting to fail and contaminate the transmission with metal which could be dangerous.
A used trannsmission is certainly a consideraton as wel as repair.
I hope this helps but let me know if you have any other questions.

I would repair it ASAP not just for safety but to prevent a catastrophic type failure. Sounds dramatic but it means a sudden and major failure which can and will damage other parts and make the repair more costly.
I would ask your mechanic to help you find a reputable salvage yard and replace the transmission with a used one. Rebuilding a tranny can take special tools and parts are expensive. Finding a good used one that your mechanic can inspect prior to install is the best way to go. Try to get the flywheel and cluych with the transsmission. I reccomend this because it is a good indicator of the transmissions condition. Also, you can use the better of the 2 flywheels, yoursor the salvage, to resurface as best possible. Flywheels are tough these days as they are generally a replacement item with the clutch. Olser cars were thick enough to re-surface. Let your mechanich inspect them and I would take some wet/dry or emery paper and cross-hatch the surface, that will generally get you by.
I would replace the following with new parts; axle seals and the shift shaft seal, the clutch friction and pressure plate, pilot and throwout bearings and consider replacing the axles as these are hard items to service and will be out to replace the transmission.
Let me know if I can be of any further help.

Glad I could help. Let me know how things go, it is always nice to know everything comes together.

Take Care.

Dr. C