New Mexico allocates 22% of elk tags for a particular unit to non-resident; 12% of the total number of tags to those nonresidents who hunt with an outfitter, and 10% to those nonresidents who are DIY'ers. Many units in northern NM are relatively easy to draw and have tons of elk. Finding a 350" bull there would be difficult, finding a 300" bull would not. The units in the Gila seem to be harder to draw, but there are some bruisers in there. This is where I've done all my NM elk hunting. There are units like 21, 22, and 23 that are real easy to draw, and hold some huge bulls. The thing is that the elk densities are low, and the terrain can kick your a$$. To shoot one of these bulls, you have to know what you are doing, and hunt like a man. NM has no preference/bonus point system, so annually you start on an equal foot with all other nonresidents.
In Arizona, you could technically draw an elk tag on your first attempt, but I know guys who have not drawn in 10+ years. Then again, I know several people who have drawn on their first try. A guy I work with drew three years in a row for a really good unit. If you get drawn in AZ, you could potentially shoot a monster with little effort. I'm not saying it's a cake-walk, but with the very low number of tags given out each year, and the high densities of elk, and high number of mature bulls, it would not be too difficult, in my opinion, to meet your 330" objective. The topography in most of the best elk units is fairly gentle also. The hardest part will be to draw the tag.