Record player sounds out of tune and "wavy"

@timattin 

Several things come to mind here after watching the video.

It appears that the particular record you are playing is not concentric (the arm is moving back and forth).  This could cause the warbling your referring to while playing THIS album.

It could be a belt.  This is an easy thing to inspect.  Remove the mat and you should have a good visual on the belt.  it should take a little effort to stretch it.  If not, replacements are cheap and easy to install yourself.  IF the belt is good, it would point to speed control issues which would take a qualified tech to address.  Probably decision time at this point to repair vs replace?

This turntable uses a P-mount cartridges.  Good ones, an improvement over OEM, can be had for around $99. The stylus should be gentle on your record collection.  This would make sense only if your speed control issues are resolved.

Good luck.

+1 for the Fluance RT82 recommendation for your $300 max price range. Also look at U-Turn Orbit Turntables in that price range.  While a belt will be cheap for this problem child turntable, that might not be the problem - but still worth a try. It does sound more like the motor itself is giving up the ghost or the speed control circuit is not doing its job anymore. Nothing last forever, especially fairly inexpensive turntables from the 1990s.

If you CAN fix it, Audio Technica makes a cheap P-Mount replacement cartridge, but better Grado Prestige models are available in P-Mount. 

I have an old Technics P-Mount TT still going strong (knock on wood) and use it to play used (flea market and Discogs) albums before trusting them to my better TT. 

I also had a slipping mat problem.  I could stop the record spinning with slight finger pressure on the side of the record.  Slipped both record on the mat and mat on the platter.  I washed the mat and wiped down the platter.

Could be a worn stretched belt, motor, motor speed control, bearings, anything on the spin part of the turntable.  I don't think arm cartridge will cause this.

There is a Rega Planar 1 demo listed here for $113.  Oooo, 5 hours left.

The Fluence and Rega are good suggestions, but better to stretch for the Project Debut Carbon which includes a cart as it is a much better deal/value.

Clean the belt with isopropyl alcohol and the platter with brake cleaning fluid. Make sure the belt is positioned correctly and the set screw on the motor's pulley is tight. If this does not fix the problem you have a defective turntable.