Amother 2cents worth,
As above, that should have been the first step. If the spindle has been damaged by spinning it was most likely due to have an arbor fitted without retention i.e. a drawbar through the spindle.
You have stated the lathe is old and worn, it can still be set up to work well and problems slowly worked through. A first step would be to level. I have 2 expensive levels but this method:
is cheap easy and very accurate.
After this apply 'Rollie's Dad's Method' to check headstock and bed alignment.
Align tailstock to headstock.
If you find you need to grind internal taper hold a piece of scrap in the chuck. Turn a 60 degree taper on it, fit a center to the tailstock and hold a new or good condition arbor with the correct taper between the two centers as per Bob Korves. Set you compoud to this,(taper), surface, fit your newly made/cobbled together toolpost grinder, true wheel and proceed to make very light cuts checking frequently with your good arbor and dye.
Don't rush and double check every step. Make notes as you go and you will end up with a good lathe that you can take pride in.