Sorry to hear your stone is not smudge free yet. Even though you will likely be sending the ring to DK for additional reasons, there is a product I want to recommend to you. I use it all the time and I think it may remove the adhesive from your ring even if it is on the bottom in hard to reach places.
It is called De-Solv-it and it has removed any and every sticky thing I have ever tried it on, including dried on paint, mineral deposit stains, candle wax and label adhesives from plastic containers. It works as well or better than goo gone but is a gentle, bio-degradable citrus based solution and has no harsh chemicals. It is a bit oily so the item needs cleaning with soap and water afterwards.
I swear by this stuff and have a spray bottle in every bathroom and my kitchen. I usually buy it at Wal-mart but you may also find it at BBB, or your local Ace or other hardware store. Here is a link to the product website:
http://www.de-solv-it.com