Thank you all for the replies!
I tried playing in different locations as Devil's Dream suggested. The ringing is less noticeable if there is additional ambient noise. Playing in the closet sounded about the same as an open, quiet room.
I have a set of earplugs I can wear that, while not eliminating the ringing, seems to improve the ratio of "intended note" vs "additional ringing".
Regarding tinnitus, I don't notice ringing in my ears when not playing my concertina - however, maybe it's there and I've just grown used to it. I'll plan to schedule an appointme