I love dawn, it tackles anything greasy!!! But if you don't want to bathe her, how about brushing through her fur with corn starch/baby powder, that absorbs grease too.
My pups get coconut oil for fur and for boosting thyroid, I also give them sea veggies sprinkled on their food (it was in one of the dog cookbooks I found and I asked my Vet about it, he said it was a great idea, it contains dulse seaweed which is an EXCELLENT thyroid booster, they love it. I also supplement with cod liver oil for dry skin. At high altitude dry skin for all of us is a constant concern it helps. Anything from the Omega family is fantastic for helping the thyroid. Diet is the key to health for us and our pets!