Another option is to look at storing them without the jewel cases, just in sleeves. A lot of upfront work, and the cost is a smaller cabinet along with a whole bunch of sleeves that can store/display the covers. Digipaks, box sets, and special format CD's are the exception. I investigated this some years ago, got a fair amount that I'd like to keep but save some space.
And yet another option is to just get several binders with pages/sleeves that can hold the CD and cover/insert.
But bottom line, I'd recommend losing the jewel cases.....and to Doug's point, sell off or donate what you don't really want.
Now, if only I could take this advice....!