Hi, I hear you on the worried bit, we moved to MT in early spring (still lots of snow) from the east coast, and I was worried, too.
Where are you moving from-to? if at all possible, I would take the southern route, as in through arizona! But one thing my dh did, was if the weather was bad, he would just stop and get a hotel. Planning out your route is really important, I think triple A could help in terms of where to stay, and weather, and road conditions. You can get a personalized map from them, tracking your route. Our friends did this for us and it was a huge help to us in the trip.