1960s stereogram.bsr turntable turning to do fast.

If I can understand you right, the turntable is running fast. I don't think you need a new motor, generally the deck will run slow if the motor is knackered (or not at all). What you need to do is remove the platter, you will find a horseshoe bracket holding it on at the spindle (it may be covered up so you wil have to remove the cover part often a silver disc). The most likely cause will be the rubber idler tyre being worn damaged etc or out of place. There could be grease dirt or something else causing the problem.

The problem is when these turntables were manufactured they were not checked and all ran fast in pitch. about 5% fast.
its a pretty easy repair but you will need a record strobe for checking the speed you can purchase it on ebay for approx. $20.00. remove the e clip that holds the platter on and when you change the speed control you will see the ider tire move to different positions on the motor spindle. take some sandpaper and turn the motor on and rub the sandpaper at the one spot for the speed you are doing, do little bit at a time and keep rechecking with the strobe you will see it will slow down and when the striobe lines look like they are not moving in either direction you are correct on the speed.do it for each speed.
this remedy works great on the early1970's panasonic stereo compact systems.