Can I use dry concrete mix to set a post in saturated soil?
You should not put dry concrete mix into a hole.   The wetness is really not the big thing it is the fact that you may not mix it properly meaning you will have pockets of dust as things settle.   These small pockets of unmixed mix will make the concrete extremely weak and will crack or crumble apart - "a buddy had this issue one time" haha.   

You mix the concrete as normal (maybe shy on the side of drier mix) and simply dump it in the hole. It will displace the water since it is heavier. I have also dammed up holes and pumped out the water while filling - this just takes a few minutes a hole and keeps the hole integrity better and keeps from getting concrete slopped everywhere.   

For wet conditions I use Quickcrete fast setting. It won't cure/set in 40 mins under water but it does harden enough to hold a post and move on.   Others may have personal favorites for wet conditions but I can say that using the right type is a big deal.