Record player sounds out of tune and "wavy"

Sounds like a speed control issue. As mentioned above, any investment in repair would be better spent toward a new table. Look for a used Rega RP3

The most likely cause is an old stretched belt.  Cheap fix--if you can fined a replacement of the correct size.

For $300 go get a Fluance RT 82 and don’t look back. That table likely has a speed stability issue and isn’t worth fixing. There are other tables in the $300 range but the Fluance is as solid as you will get at $300. 

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

Speed control. A new belt will probably help. There should be an adjustable speed control on the motor. There are apps like RPM for the iPhone that measure RPM and wow/flutter. 
 

Good luck!