I hate putting lotion on too. I hate the greasy feeling, I hate how it gets under my nails. The most popular generic Christmas gift for women is scented lotion, and all of mine goes straight to the nearest women's shelter!
I've found that Udderly Smooth (meant for cows, comes in a red/white/black package) absorbs much better than most lotions...has almost no smell, and works REALLY well. It forms kind of a coating so you won't have to put it on as often. You can see it on your hand for about 2 minutes until it absorbs, but it's more solid rather than being greasy.
I always wore neoprene or vinyl gloves when I had to apply lotion to a patient. You might try wearing gloves when you apply one of the high quality lotions other posters have recommended to your elbows. Elbow skin doesn't have a whole lot of nerve endings, so if you avoid getting it on your hands, it might be easier for you.
You could also apply the lotion to a large band-aid or gauze pad, then you wouldn't have to touch it at all with your fingers.
Also, you might want to just treat your elbows at night with something like Vaseline. You'd smear it on, then cover it with part of a sock (cut the toe out of the sock so you can pull it all the way over your arm) so it doesn't get everywhere. Then in the morning, wash it off.
I've also started using olive oil soap, which greatly helps with skin dryness and acne problems. It doesn't overdry your skin like other soaps (Dial, Zest, etc.) do.
Good luck!