Good luck with the new laser, only one of mine actually needed it. If this doesn't fix it, your real problem will be with all of the pots on the top of the laser control board. Be sure to mark their exact positions before attempting any adjustment. The pot functions are described on this page dealing with older CD players:
http://www.ralph-toman.de/cd_player_...player_eng.htm
T OFF (Track offset) was my culprit. Since the adjustment, it's been very solid. I even used it as a portable CD player last night, if you could believe it. It's an early CD player though, the kind that only likes to sit on solid non-moving surface (so don't turn it over while playing).
I'm on my second CDX bought broken to repair (for my personal enjoyment). Anyhow, this 2nd broken unit had lots of issues. Since you are new at this, be sure NOT to lose the small spring that pushes the laser track into the motor gears. Also, be prepared to replace a power cap or two. A lot of times people do not know any better and use and under-rated Gen 2 AC adapter with it until it burns things out. If any caps are vented, swap them out while you are in there.
And...round of applause...for trying to save another one of these rare systems. I have a particular soft spot for them. If I buy a third, though, I think I have some sort of weird Sega CDX repair obsession