If funds are tight, stay in the Lower-48 or nearish the border in Canada. AK can be done on a reasonable budget, but it's still probably going to be more than anywhere down there due to the high cost of transportation, food and lodging, etc. The weather can be pretty ugly too, if you pick the wrong time to come up. Down there, flights are cheap(er), driving is a reasonable possibility from most parts of the country, and you can watch the weather as the season goes on and figure out where you want to go on more of a spur-of-the-moment basis.
I second the snowcat option, as well. I don't know if I'd plan a whole trip around it, but maybe find a good resort with some lift-accessible backcountry stuff (assuming you have some idea how to read terrain and avoid avalanches) and do some cat runs while you're there.