|There are some threads on this forum about people's preferences for how to deal with diapers at a caregiver's/daycare that you might want to search, just for different options.
When I was out and had a dirty, I always kind of rolled it up to keep the poop inside that diaper and not all over everything else. Eventually I started carrying bread bags and just rolling and sticking the dirty diaper roll into the bag and knotting it, then dealing with it at all later. You will likely actually find that old poop is a little easier to get off the diaper than fresh, so cleaning these up isn't too bad. If you find that things are getting too messy in the wet bag or that the poopy diapers are too smelly, maybe you could consider having those bagged before being put into the wet bag?
Other moms who've done this with caregivers may have better advice than I do on what specifically works well for them.