If you asked my wife, the answer would be that you use them for a day or two tops, but the important part is that you throw them in the hamper wet, and then make sure to put other clothes and stuff on top of them so they sit there damp and mouldering until laundry day comes around.
Our towel bar is directly above the heating grate, so towels, properly hung, will dry fairly quickly there. Considering that towels are typically only used to dry you once you've just thoroughly cleaned yourself, they won't smell like much of anything but maybe soap and shampoo for many days of use, assuming they are able to dry out. But apparently it's more of a priority that they get put in the laundry basket immediately, moisture be damned. I gave up trying to fight that fight long ago.
You don't have a cat.