Some cats never do, but here's a couple of things that may help.
Clumping litter is usually good. If you use the scented stuff, as weird as it sounds, switch to unscented. In the enclosed litter box, the scent may bother her and whe may just be trying to get out as soon as possible. (Cats have very sensitive smell, Just because humans like a scent doesn't mean the cat does. Ever been in an overscented bathroom you wanted to get out of quickly? Same thing.)
Personally, I love those enclosed litter boxes. Unfortunately, most of my cats don't. So I've got deep litter pans and the covers ended up out on my enclosed porch. Perhaps she feels that since the box is enclosed, she doesn't have to cover, it's 'buried' already. You could try removing the lid and seeing if that helps, particularly with unscented litter.
If she's still a kitten, you can try following her to the litter box and gently moving her paws in a burying motion after she's done--often they get the idea and pick it up on their own. If that happens, praise her every time she uses the litter box to reinforce the idea.
As for stinky poops...Cat poop does stink, but an upgrade in diet may help. Higher quality food means less filler and more digestion, so less cr** in, less cr** out, if you get my drift. And a different food may smell better on the way out.
If none of that works, you could get an automatic litter box, like Littermaid. Since it rakes out the box every time into an enclosed container, this should help the smell. There's also one out there that rinses the box--it's connected (easily) to your house/apt water supply.
Hope that helps. Don't send your new friend back to the shelter. If they're not no-kill, it literally could be a death sentence for her. With a little effort and patience, you can definitely work this out.