I would never recommend going to a new, unaccredited law school with no alumni base in this economy unless you are paying nothing or almost nothing for it.
University administrators will make a big deal about a school being public, claiming that it means they are somehow better positioned to serve the interests of their students, their communities, the state, etc. Don't buy it for one second. There's no meaningful distinction between public and private law schools. This one, like its peers, will exist primarily as a vehicle for funneling a vast new pool of easy student loan money into the university's coffers.
What if they do become accredited? Does that change the playing field? How long does it usually take for a Law School to become accredited after just opening its doors for the first time?
Doesn't really change the calculus at all. The emphasis really is on the fact that it's brand new. ABA accreditation is pretty much a rubber stamp and it's essentially a foregone conclusion that any school affiliated with a marginally credible university will end up getting accredited.
The reason the newness is important is because low-ranked, local schools (which this school will be) are usually a bad idea, except for those that have a solid niche they occupy. These are typically schools in rural areas and/or low-population states, who face little competition for jobs from more prestigious schools. This means that UNT is extremely unlikely to be one of those low-ranked schools that is a good bet; it will be at the bottom of the food chain, because Dallas is a desirable place to work and grads from UT, SMU, T14 schools, and probably a number of other schools in the region will be first in line. The fact that the school is brand new means it has definitely not achieved any niche that would make it worth attending. (There are arguable cases of worthwhile low-ranked schools even in crowded markets—e.g., CUNY, which is very cheap and completely public interest-focused—but they're rare.)
That said, at some level of generality a JD is a JD, which is why I think it might not be a completely god-awful idea to attend this school if 1) it's free, 2) you want to stay in Dallas, 3) you don't expect to actually get a job practicing law, and 4) hopefully you can go part-time while maintaining your current job. That's a lot of caveats. I'd really only recommend it to someone for whom having the JD credential itself is all that's important—a person in a civil service title that needs it for a promotion, maybe, or someone who can join their dad's firm—or someone who just wants to study law for some sick reason but won't care if they can't get a job.