If you can see any rollers when you remove the paper tray, clean them -- damp a cloth with a gentle solvent (alcohol, dilute washing up detergent, isopropanol etc.)
You need to replace the pickup rollers and probably the separator pad. I can't picute the LJ III system but they usually have a separator pad on the paper tray middle top, and a roller right over the tray.
The issue is that I don't seem to find any LJ III parts for sale. You can clean them as fihart suggested, but the problem is probably that they are very worn and cleaning won't help there. You may get it working for a few pages but it will mess up again.
Assuming the issue is not with the motor or gears, you did not say if the printer was grinding or anything.
Every time I try to print something I get a error 13 paper jam, but it's not picking up the paper. Help
Have you opened it up to check the paper path? Fuser? Sensors? Does the manual feed tray work?
It's a very old printer, we can't fix hardware issues online, which is what this probably is unless you are just missing a peice of paper stuck somewhere.