You do want to get the antenna out away from the house as far as possible - unfortunately wall warts are inherently noisy, because they're generally nothing more than cheap transformers with virtually no shielding around them, so using batteries is indeed a better bet.
There is a certain amount of shielding you can do - wrapping any loose wiring to the wall wart in a ferrite core is one option - but seeing that you're dealing with a portable (which itself doesn't have a lot of shielding to begin with), it may help some, but how much is very much a product of your environment.
As to the antenna - folks on the East Coast have to be more careful, as many broadcasters - they're getting fewer and fewer - beam directly this way. Not so in NM - so a longer antenna may actually be usable, as long as you don't get so long that you start hearing your local MW station on HF, which is a pretty decent sign of overloading. There are limits, of course, to just how long to make an antenna before it has results that you might not expect, even using a better desktop radio.
Loops are always a decent choice - there are several out there, and there are plans for making your own. We have an article in our wiki (all links are blue) that has links to several of these...
Loops - The RadioReference Wiki
We have a separate receive antennas
forum - feel free to ask questions about potential antennas for your radio there.