I don't know the unit all that well but have hunted in surrounding units multiple times. You do have a good tag but deer densities are not high in the unit. For coues I would look at the areas with higher elevation. The good thing about your tag is there isn't too many other hunters so you may see a few hunters but if you spike out and camp for a couple days, I wouldn't be surprised if you didn't see any other hunters. My guess is there are a fair number of coues bucks that die of old age in 27. If you don't find a big one, I think you have a very good chance of finding an older buck if you put in the work. I doubt there will be any significant rutting activity until the end of December so most bucks will still be in bachelor groups or by themselves.
As far as illegal crossing activity, I wouldn't be worried. You probably won't see any illegals but you may. I've spent a lot of time the last few years calling coyotes in the desert along the NM/Mexico board and rarely see any illegals. Just be aware of your surroundings. Most of the "coyotes" crossing the boarder illegally want to avoid you just as much as the coyotes that hunt rabbits in the desert. East of unit 27 there is currently a fair amount of construction of "the wall". I wouldn't be surprised if the wall construction was actively taking place along the southern boarder of your unit during the hunt.
Have a good hunt and don't forget to come back and post some pictures regardless of your success.