I don't take a scrubby with a sponge attached, but use a plastic one that is more like a chore-boy, just a plastic mesh thing.  That and an SOS if we are cooking over a fire (for the sooty pots) is all I use for washing dishes.

I do take a small bottle of dish soap, but it isn't drippy and have never had a problem with it.

After scrubbing dishes the scrubby is rinsed with hot water (recently boiled) like the dishes and packed into a ziplock bag wet.  It never gets unpleasant and I have never worried about bacteria.  However, if I were to do so (and now I may) I would use the "few drops of bleach) method.  But I would not take a scrubby with a sponge attached, ever.

After a trip, a used one just is thrown away, and of course so are any used SOS pads.  I use the SOS very sparingly and often only use one on an entire trip.

These days dishwashing liquid is antibacterial so the sponge tends to last longer than it use to IMO. My "silverware" ... cups and stuff (toothbrush) hang from a tree in a mesh bag but the soap and sponge go elsewhere in an open upside-down bag so the sponge can dry, somewhat. (I can dream at least.) In any case, I don't want that sponge anywhere near my toothbrush.

I have always used a water bottle cozy that I'd keep wet in warmer weather. Now with all summer trips it's always wet and bacteria poop will normally noticeably sour it in a few days. I don't expect it will kill me either but I can only ignore it for so long so I've gone the bleach route by repurposing a little, maybe 1/4 oz bottle that once held eye drop something or other I noticed in the recycle bin one day. I've been watching for a nice bottle twice as big for soap as that would last me two weeks easily.

I'll throw the sponge in the bleach bath too just cuz I can I guess. It's still not going near my toothbrush.

I am amazed the cozy has retained its color over the years. I was actually hoping it would get a few shades lighter by now but you only need a drop or two per 1/2 quart H2O to fix the smell.

If I remember correctly, bars and restraints use some sort of sanitizer that I expect might work too. Don't know if there's any advantages or not. Just a possible option if available.

LMAO

Marlin

Now they make TP without the dreaded, ultra-heavy & bulky cardboard tubes.