Hm. Now that you said that actually, let me think here.

Okay, I can heavily modify this whilst keeping the location of Phoenix and the surround urban area known as the Valley. I don't think this concept could reasonably stretch out beyond the immediate outlying areas of Phoenix. It'd practically be a severely more militant force born of survival. They'd still have a hefty amount of Legion indoctrination branded in their minds, with some key changes. With their situation so dire following excessive infighting and massacres, women, the elderly and even children are seen picking up arms against aggressors. A sort of forced concept of alienation and the embedded concept that every able body must fight to protect their own. Sadly, this way of life has led more than a few women of any age to be forced upon by some of the more renowned warriors. Pregnant women would be a more common sight than anything else walking through certain settlements within Phoenix. Slaves would be more common as well as commonly sought and treated like cattle just the same, nothing more. Just like the old ways, however with women (albeit very slightly) progressively reaching equality through recognition in battle certain customs have changed over the years. Although whose to say if they don't quickly rise to a significant fighting force and banish women back to their previous role in Legion life and swiftly return to the old ways. Some key female warriors may just get things to gradually change if they live to make such changes.

Now, i'm having trouble thinking of how this nation would lead itself. My concept as described is that this new legion is all that's left after such infighting. So few legionnares remained, and even a handful of female and child slaves and children training to be legionnares survived after several merciless slaughters of slaves and children. A single high ranking Vexilliarus attempts to proclaim himself the new sovereign leader, he is cut down by his own men. No one wants to continue this endless killing and succession, so i'm trying to think if they'd attempt something like a senate or if they would have something like a council of legates in charge, or just promote a sovereign authority figure.

Edit: Aside from this, I also have the idea in mind of creating a rogue faction of brotherhood of steel that's transformed into some demented technological cult revering the power of tesla energy as some omnipotent entity and thus hoarding and envying the most advanced technology they can get their hands on. Basically, the greediest Brotherhood sect to date, and the craziest. They'd have commanders be called "Inquisitors" and go on regular scouting missions sending both Inquisitors and Missionaries to discover "heretics", practically anyone in possession of the technology they require which includes other Brotherhood sects, or "the blind", potential wastelanders or settlers they believe could be easily coerced or manipulated into joining them and their Tesla Crusade. They'd just inhabit a single vault or series of bunkers somewhere in the midwest, and if I remember researching correctly radioactive tornado season would probably keep them locked up in their vault or bunkers.