Ben,
No real info about Wyoming because I have only either driven through or visited Devil's Tower (which was incredible btw). But I have a few thoughts.
Carl brought up a good point about the family structure and it being tough on kids (which you already recognized) but I will take it one step further. If you rely on your family to help watch the kids during the day or whatever, it will also be tough on you. You may be looking at increased child care costs for you and your wife if you move away from the family structure you have, so something to consider if you have not already considered that aspect of it.
If Wyoming is some place that you really think you might want to live, then my advice would be to take a couple of vacations to the state over the next few years and stay there during the WORST times of the year. The height of fire season, the deepest depths of winter for example. See if you or your family can handle that. Fires can be bad out west, and they will get worse before they get better I am sure. If you or anyone in your family is prone to allergies or breathing related problems, that may be something you need to consider. You guys might find a love of cold weather sports by going in the dead of winter. I mention that idea though because I have friends that have visited places during the nice parts of the year for whatever location they were at and they end up moving and reality hits them and the nice parts of the year are only that way maybe 20% of the time and the rest of the time is miserable. So, visiting during the awful time of year will give you a better idea of what it might be like living there year round.
Having visited Eastern Montana multiple times, I can tell you that the wind can be incessant on the prairie. I'd agree with a lot of what Brad said about Montana. I deal with hurricanes and tropical storms as much as you do but I would say that the wind up in MT is way worse than dealing with the wind down here. It can blow for days and it gets old quick. I was never in MT in the really cold parts of the year but there were days where it blew easily 30 mph for days, with stronger gusts and it makes hunting very tough and sometimes miserable when you are cold and can't get warm til you get back to the truck. Add in the cold and I'll take a warm, humid hurricane over cold, incessant hurricane or tropical storm force winds for days any day of the week. I kept an eye over the winter on my sage grouse areas and the last few storms negative temps and constant 25-35 mph winds has been the norm. There is so much about Montana that I love, but I really don't think I could handle the winters there. Snow and cold might be something to get used to but I don't know that I would ENJOY getting used to cold and wind and wind and more wind and even more wind on top of the cold (with or without snow).
I didn't know that WY was the highest rates of suicide per capita. That is interesting. Something else to consider is looking at drug use rates. One thing that I have noticed in all of the rural areas of MT that I have visited is the numbers of Meth Kills types signs that are posted. Petroleum County is one of the least populated counties in the lower 48 and the number of Meth Kills type signs is amazing. I talked with the Sheriff about it and drug use is by far the biggest "problem" (I think domestic violence came in second) for law breaking. So, wherever you move you might consider looking at the things that kids can do outside of school and what kind of social network they might have. Not gonna say that your kids will ever have a drug problem, but depending on where/if you end up relocating, they might have a higher chance of dealing with drug use by peers than where you live now.
I haven't hunted WY yet so I can offer no real information that wasn't already mentioned. I will say that Eastern MT ducks and geese seem to be more patchy and here today, gone tomorrow. I say that though knowing I never really targeted ducks or geese while we were out there. There were always a few ducks on potholes and stock ponds that could have been shot if I had gotten the migratory bird stamps and what not but I never did since we were out chasing upland game birds. There were a couple times that we were going to go shoot a pond or two for ducks but they weren't there the next day or the day after. If WY is that way, then you may not be thrilled with the duck hunting opportunities. But upland birds may easily take their place. I remember one year was an incredibly bad drought up in MT and still we made the trek to hunt it. One rancher asked why we made the trip in such a sucky year and I responded that even a bad year in MT is way better than a good year upland hunting in FL. I imagine that it would be the same from someone from NC....lots more hunting opportunities period out west than in the east. So, I imagine you would find something to replace ducks quite easily.
Access laws were mentioned for fishing, you may want to look at the same for hunting access. MT has great programs that allow hunters to access private property. WY might as well.
Those are just a couple of thoughts that I had....
Dani