You probably have a PSC or a CSCR motor.
PSC (Permanent Split Capacitor) motors often have the replaceable cap under a bulge like cover attached to the outside of the motor.
CSCR (Capacitor Start Capacitor Run) motors will have two of those bulges. One for the run cap and one for the start cap. When the motor is stopped the internal centrifugal switch should be closed so the start cap is in circuit. If it is stuck in the open position and if the capacitor does not have a bleed resistor on it then the capacitor may remain charged and is a SHOCK HAZARD. I’m making that bold because it is a safety issue to be aware of. You can use an ohm meter to check for continuity once the capacitors are discharged and removed, to check the switch and windings.
If your motor has an open winging this can allow one or both capacitors to remain charged and present a SHOCK HAZARD.
Capacitors fail often. They are replaceable for a reason. Just be careful when you do it to discharge the capacitor first and replace it with like for like.
I suspect with a motor that old you have worn bushings. Usually not worth fixing for a small band saw type motor. End to end play is normal for bushing type motors (ball beating motors do not have end to end play), but side to side play is the killer. Disconnect the drive from the shaft and give it a firm push and pull in all directions. If you feel nothing that’s good. If you feel a tiny click click then your bushings are worn and your rotor is probably contacting your stator when the magnetic field energies, Time for a new motor.
Your bushings May also have dried up. Those motors typically have oil ports for servicing. If they’ve gone dry and this is your telling sign then the damage is already done and you will waste your time trying to fix it (ask me how I know)... so many unsuccessful attempts...
While your drive is off, grip the shaft and pull while spinning it. Is it smooth? If not then you have bearing damage or debris lodged causing a restriction. Fix or replace based on your findings.
Good luck, be safe.