I would like to return an engagement ring and get as much money back
for it as possible (although the ring isn't damaged, the jeweler will
only provide an in-store credit, which I don't want). Do you know of
any reputable establishments that buy back engagement rings for cash?
I'm NOT interested in auctioning it off on eBay or other auction-type
sites, or taking it to a pawn shop. Many thanks.

Maybe a small jewelery store will take it on commission.

The problem is that the market for used diamonds is nearly nonexistent
and accounts for less than ten percent of sales. I am sorry to say
that truly the best way to get top dollar for the ring is to use eBay,
as it is a highly efficient marketplace.