Record player sounds out of tune and "wavy"

I also had a slipping mat problem.  I could stop the record spinning with slight finger pressure on the side of the record. 

This is how slip mats are supposed to work...hence the name.  

OP, even if your record collection is significant, given your budget, maybe invest in streaming until you have funds to get a better TT. You could probably get a used Bluesound NODE for $300. Add Quobuz for $11/mth and have access to almost anything at CD quality or better.

I enjoy the heck out of my Bluesound and have a pretty decent analogue source, but the value for money proposition of digital via Bluesound is really tough to beat.  

I rather enjoy it when brake cleaner intrudes on audiophilia.

It's also invaluable for cleaning polymerised oil off an induction cooktop.

And it can be used on cars, including the NA Miata I bought in 1991.

It seems to be the common factor in many of my obsessions, though I have not yet learned how to use it in the development of film.....

I just listened to six or eight of these tables on Youtube and they all have some kind of speed issue.  I don't think a new belt would help much.  You want to keep it at or under three hundred bucks, used could be the way to go,  Tables like the Pioneer PL-512 or 514 (avoid 5-16,18 models) have large asynchronous motors that last a lifetime if maintained and are noticeably better than a lot of the newer so called affordable products out there.  They keep very good speed and I used to see them at $150-75. That leaves you a decent amount for a cartridge.  Dual turntables are nice also but they're more finicky to maintain.  Also check the better mail order outfits like Music direct and Upscale Audio for deals.  Anything they sell will be better by a substantial margin than what you have now.

If a new belt is not affordable, I believe the belt can be boiled in water for a certain amount of time to shrink it back to size. Look under vinyl engine. If memory serves there are posts there regarding the process.