I use Americas Storage in Texas City (really closer to Dickinson) on FM 646.
$125 a month including 120v electric, covered. The fully enclosed, with electric are around $200. New security system with great cameras. I believe they still have 24 hour security. Always someone there in the day time and security personnel at night. That would be about a 40 mile drive so you may not really want that distance. There is another one on the same road, but I don't know what the prices are.
4600 Fm 646 Rd N
Dickinson, TX 77539
I just brought up a map of Sugarland and populated it with "RV storage" and there are a lot in that area.