Hi Izzy,
You need to think very carefully about this, once you remove material, thats it, you cannot replace it !
When the lathe was made originally, the bearings and caps were deliberately made with shims, not because they were cut wrongly, but because things wear. The shims are there so that the spindle clearances can be adjusted to account for the inevitable wear that will take place over time.
The shims are usually, but not always, a stack of very thin ones, often only a thou or so thick, they can seem like one solid shim because they are compressed by the cap and tend to bond together.
The normal way to adjust the bearings is to remove one or two shims at a time, measuring the spindle rotation as you go.
When the spindle becomes more than hand tight, put one shim back, and check again. It is not a quick process ! I've seen engineers take days to shim shaft bearings. This is why I suggested feeler gauges earlier, in case the shim was solid and not a stack. Even compressing out the surface oil between the shims will affect the clearance.
Whilst I think about it ! Carefully clean the oil drain hole in the bottom bearing. The oil must have somewhere to drain. Remember any dirt that gets into that top oil cup will get into the bearing and cause the scoring that you talk about on the shaft, and lack of oil is even worse because particles of the bearing itself will cause scoring.
When you put it all back together use plenty of clean oil, 10 or 20 SAE motor oil is perfectly fine. If perchance you use multigrade, oil, do not mix it or allow it to mix with detergent oils. It will turn into a paste that won't lubricate anything.
I'll come back later and discuss thrust washers with you.