Replacement shower heads are cheap - it's not worth the aggravation to try to fix it.

Either find a replacement you like, or call the manufacturer to see whether there is a warranty they will honor. Some companies have limited lifetime warranties and I have twice gotten free replacements of plumbing fixtures that were 10-15 years old.

It is theoretically under warranty - I assumed it would be more trouble than it's worth to try that route, but maybe I will give them a call. Thanks!

Depends on the quality of the customer service department, but it shouldn't take much effort to see. A couple of years ago I had a Moen kitchen faucet that developed a leak. It took a 5 minute phone call and an emailed photo of the faucet so they could match it to the correct model. I didn't need to furnish a receipt or anything else. The new faucet arrived at my house a couple of days later.

Remove it, and soak in CLR - a possible fix.