Hello Dan!
1) The resistor posted upstream is the bleed off resistor for the lamp socket- R1 in the circuit diagram. It drops the voltage sent to the lamp socket to lower the brightness of the lamp.
2) Have you pulled the motor apart? Remove the two nuts and bolts going through the frame and remove the two armature bearing end supports. Pull the support off the long extension side so that the other side will stay attached when you pull the armature from the stator- that will allow the rubber pulley to stay on the armature spindle and not have to be removed. Once removed, check for oxidation or signs of rubbing on the armature. If you find any oxidation, take a rubber eraser and scrub the armature. If there is no galling or pitting, the rubber eraser is abrasive enough to clean the copper without hurting it. If you see signs on rubbing or galling, the armature needs to be recut and new bearings installed- realistically, more expensive than the motor is worth.
3) Do you have a multimeter? Check the resistance of the motor. If you get a low resistance, the windings are good. If you get 0, open, infinity, etc- the windings are broken and open. If you get a very high resistance, the windings are shorted at some point.
Now, as to a replacement, you have a couple options. A, you can find another projector with the same motor and put it in place, or B, you can adapt a more modern motor. You might be able to find an AC motor that fits, but I would think it would be easier to fabricate a motor mount for a DC motor. DC motors develop better torque for their size, so you should be able to find a DC motor about half the size of that one- like what is found in a VCR drive- that you can fab a mount for. From there, all you'd have to do is integrate bearing blocks into the mount with oilite or needle bearings and place the armature from the original motor in it's original place. I would turn down the armature and make it into a pulley for a rubber band drive. Given the right lathe, you could also cut a bevel gear into it and make a matching gear for the motor so you don't have any losses due to band stretch, etc.
On further thought... it would appear option B is not really much more cost effective. Unless this is a very special projector for you, or a project you'd rather die than leave incomplete very long- I know what that feels like- your best bet might be to put it on the back burner until you find an identical replacement. The difficulty isn't going to be finding a matching motor, it will be finding a matching motor with the same shaft length and size as what is existent on the current motor. Motors in general are/ were universal, but manufacturers commonly ordered a special shaft length or size to fit their application.
Hope that helps a little, Michael
Edited to add- PS- if you go with a DC motor, you'd also have to integrate a small transformer.
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