Sounds like a speed control issue. Agree with checking the belt. If it’s that old it may have a defect and as previously mentioned that is a cheap fix. If that doesn’t do it then it may be something with the motor. If the motor is the issue then getting it fixed for a reasonable value relative to the total worth of the turn table might be difficult.
I haven’t priced entry level turntables for a while but for $300 you should be able to get something decent from Pro-Ject or Audio Technica, and given the explosion of interest in turntables over the past few years a company like Sony might have something decent in that range as well. Your cartridge might be salvageable but at the minimum I would get the stylus retipped if it’s 30 years old, as you don’t want to damage records. Most entry level tables come complete with a new cartridge and I would go that route as it will probably be a step up over your old one.
Do you have a phono pre amp? Most budget turntables include a preamp but they are usually awful, quite audibly worse than budget phono preamps. Hopefully you do because a decent entry level pre amp is about equal to your budget.
Your best outcome here would be if a new belt fixes the problem. And I would get a new mat as well, because they can rot with disuse. A new mat and belt can be inexpensive as others in the thread have noted and I would try these first