1993 Toyota Corolla Right Turns Shake REAL Bad

Sounds like the drive axles (CV joints). If you can raise the car and put on jackstands, check the inner and outer protective boots. They can get torn by road debris, lose all the lubricant and then grind metal to metal.
They would need to be replaced. Remanufactured complete assemblies are about $60 each.

Remember that Toyota uses a seperate lubricant supply for the differential (Dexrton III), so be sure to refill the differential after installing the new axle(s).

I don't believe I'd have the tools to remove the axles. How much do you think it would be at a shop? Including the purchase of the CV Joints? And where would I find that seperate supply for the differential? Is it anything like the 1977 Toyota Landcrusier FJ-40 where you have to have a special "pump" to pump the grease into the axles?

The axles are held in place with a small "C" clip, they pop out.
Usually you can put a prybar between the transmission and the axle housing and pry them out.
There is a special slide hammer tool made for this procedure, you may be able to rent it.

A shop would charge between $250-400 for one or both axles and seals.

The differential housing has a plug on the rear of it on the firewall side, it is hard to get to unless you can raise the car, but can be done. After axles are in remove plug and fill until it comes out, replace bolt.
You can buy cheap plastic pumps at an auto parts stores that screw into the trans. fluid bottle, and pump it into the hole.