I have collected LOTS of charms too. I usually buy them as precuts in packs. The packs are the same size as CDs/DVDs, so I started using a tall, narrow CD holder I found at a 2nd hand vintage shop. I filled a second one that looks like a mini bookcase for CDs.
I don't use them fast enough, so many have been stored away in bins too. But I like being able to see them.