I'd try a different collet and accessory first, but if that didn't do it, I'd take it apart.
FWIW, I had the bearings go out on two BB Dremels, so the last one I bought was a sleeve bearing. Works fine. Actually, with all the dust ours are subjected to, I think sleeve bearings are a better choice.
I have an old Skill 3/8" 'lectric drill that dates from the '70's. It ran for a couple of days in Oct. '81, running the ODO back on my '81 Mazda so it was back on warranty. When driven in reverse, the ODO showed fewer miles, an oversight on Mazda's part. Just after warranty expired, I noticed the transaxle output seals were leaking. Anyway, the Skill drill bearings started squawking (twice), so I took the drill apart and oiled the bearings a couple of times during the weekend. It's still my #1 corded drill, tho I recently bought an 18v. Makita. The Makita purchase was only because I wanted to use the battery pack to run my Sullivan starter. The price of the battery and charger just made it a better deal when buying the whole kit.

Steve