Spindles are rebuilt with high precision in a CLEAN ROOM. Bridgeport used to have one. I saw it.
Spindles are greased with typically KLUBER grease and even that is a science. So many CCs here and there. Too much and you generate heat.
The oil may be coming from the transmission above it. That has a sight glass. Check that.
How does it spin on the bench? How does it feel? Believe there is a good spindle repair facility in Atlanta.