Matilf,
Do you live in town on a small lot? It sounds like heat is the only thing you want to avoid, and having relatives in Tucson, I hear 'ya.
If you have the physical space, lay a concrete pad and buy one of those canvas car shelters. You will have to replace he canvas every couple years, and extend the legs with wider pipe, but it is quick and dirty. The other option is to tarp the TC during the non-use times. I did this with a heavy, light colored (ours is sand colored) duck tarp and it is still alive after 10 years.
I actually built a 1200 sq. ft. building to house both my model trains and our TC, but misjudged the height of the roll-up door putting it at 10' to clear, whereas the TC is now, with suspension upgrades, 10'-4" to clear. The trains have chased everything else out of that building anyway.
The other option is to build an RV shelter. Many on here have done just that. It finally came to me that we had the space (2 acres) and needed a snow/weather shelter so I built one: It's 10x24'/no plumbing/no electrical/no doors/no windows/no floor (I spread some crushed rock base around)/no permits required. Basically it is considered a pole barn. The 4x6 posts are in concrete, 4 feet below the surface.

With the TC up on stands I can also park most of my Kioti tractor in there. Here pictured with the mighty rear pull snow blower, which you would never need in Tucson.

There are a lot of ways to make it work. Find the most appropriate one for you.
jefe