You have two options on this matter.
1. Get a new CD player, the new ones have a much higher over sampling rate than a 20 year old one. That prevents CD skipping.
2. Get a small mpeg player, it has no moving parts and you can store more music into it. I use a little Sansa Player that only costs around $30 and it doesn't change much of the music quality.
I've converted all my CD's to mpeg and I'm very pleased with it. The reason why these CD's skip is due to the wishbone suspension design that holds the Lazar eye in-place and it takes very little vibration to cause the skips. I don't think that even one with a high over sampling rate would do much justice. I have the same problem with the CD player in my car and I use the cassette adapter with my Sansa after paying $300 to replace the radio unit. Also I worked for Ford Motor Company Electronics Division repairing these things. There were lots of complaints from the customers back in the days about this matter. I use Foobar2000 software to convert them and it's free online to download.
So invest on this little item and get rid of this headache before all the prices go up due to what happened in Japan lately.
I'll ask AndyB if he might be interest in my old 5 disc player, he might be interested in yours also.