I'll bite.... I've been in a few caves within the area you specified. Blowing Cave out near the now imploded Nutty Putty cave was nice. It does have a vertical entrance requiring ascending skills. I don't know the current access info, however. My recommendation would be to visit a "grotto" in the area and hook up with some of the local cavers. You can find their meeting schedules here: https://caves.org/state/utah/
Thanks, I caved the Nutty Putty Caves a few times before they were closed. I'm assuming there have got to be more close by, but I just don't know where they are.