The reason conventional cucumbers are coated in wax (and English wrapped in that shrinkwrap plastic) is that cucumbers don't keep long at. all.
It could be that you just need to use them faster. Do you put them in your veggie drawer (crisper)?
Otherwise, it could be:
1. What is your fridge temperature? Buy a fridge thermostat and make sure your fridge is not warmer than 40 degrees.
2. Deep clean your fridge, especially your veggie drawer, in case there are some bad bugs in it that are making things worse.