"Normally" it's not a bad drive. That of course, depends on your perspective. This isn't exactly shaping up to be a "normal" winter. While those of us who live here might not feel particularly troubled with a drive in snow and ice, if you are coming from South Texas, I would buy a ticket and fly. Why rent a car if your husband has a vehicle? By the time you rent one, buy insurance (GOOD idea this time of the year), pay for fuel, and not to mention the worry and stress of driving unfamiliar roads in unfamiliar conditions, it's not worth it.
If I-80 were to be closed, you'll pay additional costs if you're stuck in Rock Springs with a rental car. There are many times that airlines will fly when the roads are closed. It is @180-190 miles from R.S. to SLC. Road conditions can change DRASTICALLY in a very short distance around here. The "twin sisters" pass between Evanston and Rock Springs can be treacherous when the weather is moderately bad. This is a pair of steep "hills" that are back-to-back, with steep drop offs on either side for the most part. Bottom line, if you start to slide, there's nowhere to go but DOWN. It gets driven successfully everyday, except when it's closed

. But does that matter if it's YOU behind the wheel when it gets "squirrely"?
I've lived in both Corpus Christi and the Rock Springs/Green River area, among others, before moving to Cheyenne. If I were making my first trip here during that time of year, I'd fly as far as possible. BUY the airline ticket, then decide to drive for next time you come back. What those of us who live here think is decent winter driving conditions, might be horrific to someone who's from an area that 60* is considered "cold"
We moved from New Mexico to Wyoming during October/November several years ago. It wasn't the ideal time to move, so I know what you're facing.