I would get a CD/DVD cleaner disk for around $7-8 and give it a try. Things do get dusty in there.
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Keeping it running is better than buying new

The battery idea is interesting. Assuming you're right and your battery is not the culprit, my next stop would be to check for corrosion at either end of the B+ cable. Mine looked pretty great till I took the nut off holding it to the main fuse box next to the strut tower. All four wires were very stiff and corroded and all of it was hiding while it was installed. That condition will cause the same low voltage issues as an actual battery problem as far as your CD player (and anything else electrical) is concerned.

Another point to think about is that any type of CD player (car, portable or home) can sometimes exhibit this skipping problem when the laser lens gets dusty. They used to sell a disk that supposedly cleaned the laser lens, but I have no experience with them. They seem a little dubious to me, but if you or someone you know has one, it certainly won't hurt to try it. I have blown mine out with compressed air. Be careful with that method though, it creates a static charge. Narrow vacuum cleaner nozzles can do this as well.

If it were me, I'd pull the radio and take whatever covers off I needed to in order to see, and blow the general dust out. Then find the laser head and just wipe it off with a lint free cloth meant for eyeglasses. Remember, if you disconnect it or the battery or B+ cable, don't forget to have the radio code first.