Do you have a duplexer installed? If so, remove and try printing from tray 3 again.
Is tray 3 a 2000 sheet feeder? Or 500?
You say you cleaned the rollers, but they may be too worn.
Replace them. Tray 3 will have four "D" rollers on a shaft, and the separation\feed roller in front, as well as the tray roller.