I cave regularly whilst wearing daily disposables. If it's a long trip then I will take a spare set in my emergency pack, along with some tissue paper so I could attempt to clean my hands. I have put one back in underground, but I had some bottled water with me so used that. I don't need to tell you about the risks of using cave water to wash your hands before sticking one in your eye! But you take your chances.
I would strongly recommend having some hand wash and a spare set in your car in case you lose one underground, otherwise you may not be able to see the types of beer on offer in the bar after! Also, driving home becomes an issue.
One thing to note is that after doing a duck, make sure you wipe your face of excess water before opening your eyes to avoid washing one out with a load of water running down your face from your helmet. And try and avoid looking up pitches if there's a lot od drips - again, I have had one washed out looking up an aven.
However in the main I have zip diddly problems