We've had to use the 4x4 on wet grass, too... dug a really nice rut in the soft turf next to the pavement where the RV went. Camp hosts were nice about it - They assured us we were not the first to dig such a divot.
I've only ever had SRW, so no idea on DRW.
As for front-vs-rear drive... as long as you know which you have, you can adjust as necessary. When front wheel drive first came out, I recall MANY cars breaking out on the front (steering) wheels and being totally lost in the snow. While we pushed right on through with our rear-wheel-drive "sled" of a station wagon. Of course, studded tires also helped...