Why does my fry-bread turn out so oily?
When making lángos, which is pretty much identical to fry-bread in every respect except the name, there are two keys to preventing an oil-logged outcome: 


First, make sure your oil is hot enough, but not too hot. You want to achieve the desired toasty brown color on the outside in precisely the same amount of time as it takes to cook the dough all the way through on the inside, and not a second longer. (Well, OK, so that's exaggerating a bit; you do have a little leeway, but you shouldn't be frying a piece of dough much past the point of doneness just to achieve the color you want; and obviously, you also don't want the outside to be burnt but the inside still doughy.)
Second, when you remove the bread from the oil, blot the excess oil. As in, don't just put it on a metal rack and expect the oil to drip off. Go ahead and use that cookie rack or what-have-you, but line it with a couple of paper towels. Or half a dozen paper towels. Or an I-don't-know-how-people-made-fry-bread-before-paper-towels quantity of paper towels. And then use a second paper towel to blot the top of the bread.


(Important: to prevent sogginess, once you've blotted the bread, remove it from the paper towels, and either serve immediately, or put it in a suitable serving pan in as close to a single layer as you can manage.)

Another aspect that people often mention is how many times you flip the bread. Just like with hamburgers on a grill, there are advocates of the "flip exactly once" method, and advocates of the "flip many times" method, with very little middle ground. I tend to flip once, just because that's less work; I don't actually know if the number of times you flip has any effect on the oiliness of the outcome.