I have an old and clunky 1941 J bridgeport (with grease fittings)
Erm, I
could be wrong here, but as I understand it, no Bridgeport vertical mill should ever be greased - always oiled, even if it has what appear to be grease nipples (they're not, their oil nipples)....
If your BP is full of grease & has been for some time, it may well be worn in ways you wouldn't expect.
You may be looking at needing a full strip-down & re-scraping to bring it all back into alignment... I must admit, I daren't check my mill in case it needs that, I wouldn't know where to begin.