I commute in Houston. Same problem and I have to adapt to different offices and situations. Key is ventilation to dry.
I always start from home with clean kit.
In one situation, where shower had towels, I carried a clean t-shirt and I left my wet jersey on the bike. (I did have one stolen so I looped lock through it.). That way the only thing I had in locker were shorts and the locker had enough ventilation.
In a place where there were no towels, I used a small sports towel. I was lucky that I found a cache where I could hang it unnoticed and it would dry.
Worst scenario I had was having to shower then take everything back out to the bike to hang it to dry, then reverse the process when heading home.
Even in humid Houston, getting wet fabrics outside will prevent mildew/stink. Should be a snap in SLC. The key is don’t let it stay wet, and the problem to solve is finding the least inconvenient way.
In one high-end office I go to, the commuters have lobbied for a drying rack and it is promised for next remodel. Though I’ll probably be dead by then, it is good to bring up the need with the building management since they have no clue that a drying rack is needed.