I did NAIGC for two years. I'll respond to your questions with my experience specifically but of course it depends on the team...
1. Options to stay with gymnastics
You could compete NAIGC, possibly join a club team but not compete if you want, ask around at local gyms to see if you could join a competitive team there, or you could go to local open gym time and play around but not compete.
You could also take your gymnastics skills and apply them to something similar like circus arts, dance, rock climbing, Crossfit type things, yoga, etc.
If you are interested, coaching gymnastics can also be a way to keep up with the sport and stay involved.
2. What are these Club teams like
Mine had two options to join, noncompetitive or competitive. Practice was every weeknight from 8:30pm to basically whenever the people in charge left (usually after midnight) but noncompetitive team members would have to leave after only 1 hr. We used the facilities of a local gym that had basically everything you would need, although some of the equipment needed to be replaced.
In order to stay on the competitive team you would have to do a certain amount of fundraising every year. You had the option to go to about 6-8 meets total throughout the year, most of them close by. Nationals is every April and the club would reimburse members for plane tickets partially depending on the budget each year.
Practices were just open gyms and there was no coach.
My experience was that it was frustrating to stagnate in my progress due to un-structured practices and no coaching, plus not having enough time to put in due to college. I had competed old L7 and had some L8 skills, but the routines I competed in NAIGC were more appropriate for L6 and I lost a lot of skills (especially on bars). I did have fun getting to choose what I wanted to put in, and not worrying about scores. Some teammates had competed L9 or NCAA, or had crazy tumbling skills from cheerleading, so I was never going to be competitive with them.
Additionally, the club was run by students and former students, and it was pretty mismanaged. Eventually I got frustrated with that, especially in terms of seeing my money go to things I didn't agree with and seeing how some of the officers treated members badly, so I quit. In the end I would say it mostly a frustrating experience, though I do miss practicing my basics. But some people get a lot out of open gym set-ups and have a lot of fun.