I am currently a nursing student, I saw this post a couple of days ago and I at the time I couldn't replied so I sent myself the link to reply later. I too suffer from this embarrassing condition and I too wondered the exact same thing you are wondering about. I came to the conclusion that that could not stop me, if I made decisions based on this...I wouldn't leave my house!
You were wondering about the patients, well you will be wearing gloves most of the time that you are touching patients so they are not even going to notice
The one thing that I do struggle with is putting on the gloves. Some times the nurse preceptors actually have to wait until I put them on, 'cause it takes forever! I usually say, 'oh, my hands get so sticky from this purell' (unfortunately the one at the hospital is not the same as you buy in the drug store, I think it has something else in it so as not to dry your hands). Most people just agree. They do look at me weird sometimes, but I just don't pay any mind to it.
I make sure that there is a box full of large gloves available, since it is easier to put on. Also, there are sinks in every patient's room (for the most part) so I make sure to wash my hands a lot, and that helps. Also, before putting on the gloves I move them around to unstick them and when inserting my hands I try to put pressure with the back of my hands to make it a little easier since that part is not so sweaty ( this a little difficult to describe). I just started doing my practicum working the night shift and I noticed that my body temp is not so high at night so there's less sweating.
I wish that I could tell you, just do this and it cures it. Unfortunately, we have to live w/this. I heard about the surgery but I am too scare to try it since one of the side effects is compensatory sweating in some unusual parts. At least is "normal" to sweat from your hands. I am considering medication for some days.
Do not let this stop you, you will be surprised how creative you can get. If this is your passion, go for it!!!