I second the recommendation to use I-40 that time of year. I use this web site to find RV parks:
RV Park reviewsIf you read a lot of the reviews on the park, you will get a feel for the park. The reviews have not let me down so far. There are links to the park's web site so you can look up further information.
You need to purchase a trailer that will not freeze up in the winter. You need tanks that are heated or enclosed in a heated space. You need to make sure there are no exposed water lines. When you get to your destination, you will need to get a large external propane tank for heat and will want to enclose the bottom of the trailer. There are some great threads on how to keep you warm in the winter. My concern is you will get a poorly insulated trailer that will be impossible to keep from freezing.