+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.
Record player sounds out of tune and "wavy"
I've been trying to diagnose this issue for a while. My records all sound slightly out of tune and "wobbly". I've tried old and brand new records and they all have the same issue. It's not an obvious thing to hear, but I can definitely hear it when I pay attention and it drives me crazy. Is it possible I set up the turntable wrong when I took it out of the box and assembled everything? Something with the belt? Or could it be a needle issue? I'm pretty much a turntable newbie, so I'm hoping there's some glaring error I'm unaware of.
I don't hear the problem when playing anything else through the speakers (Bluetooth, Spotify, movies, etc.). I've also tried placing the turntable on the ground or making sure the table's surface is completely level. Still has the same issue.
Here is a short clip to give you a sense of what I'm hearing. I hope you can hear it too! Any help is much appreciated! By the way it's a Kenwood KD-291R.
CLIP: