If you prevail in your criminal
matter, then you could talk to a civil lawyer about a false arrest
/malicious prosecution. However, like it or not, you have been charged with a criminal offense and that has to be your first priority because a conviction or a plea to anything at all to get rid of this would cut the legs off your civil suit. A conviction makes a false arrest "true" by definition.
I know you're angry, and I'm certainly not saying you should like what happened to you. But I don't expect that this case will be immediately dismissed just because you don't believe anyone had any right to arrest you. The state of New Mexico is entitled to prove otherwise. So not having heard their side of this story, I think you'd best be prepared to fight these charges, not emotionally, but with a lawyer who will not only protect your rights on this case but your civil matter elsewhere down the road. Because anything you say in court
for yourself can be used against you in both courthouses, while what your lawyer says on your behalf generally can't.
You could file a complaint
with internal affairs meanwhile to have the arresting officers investigated, another one with the brass at I-Hop corporate, and mention both in a complaint to your state attorney general's office, but as these reports also could contain admissions that can be used against you, you should not file them without showing them to your lawyer, who can tell you if they could hurt one or the other of your cases.
Your best bet would be to find an experienced litigator (trial
lawyer) in general legal practice who knows his way around criminal and civil court. It would be best to have one lawyer for both matters so that the lawyers wouldn't step on each other's toes. If you had to go with a public defender, while he would be able to handle the DWI case, he would not be able to represent you on the civil side.