Glove fit and hand sanitizer.

Hello everyone. I have a question about how to deal with difficulty getting gloves on after using hand sanitizer. I work inpatient, so hand sanitizer is used before you enter the room and after you take your gloves off to leave the room. This is repeated for every room/patient. My problem is getting gloves on after using so much hand sanitizer. Can't wash my hands after each patient due to time constraints. The size medium gloves I can get on, but those gloves don't fit as snug as recommended. I am currently in training since this job is new for me, and my trainers want me to wear size small because they fit snugly. But once I've used hand sanitizer a couple of times, it's impossible to get those size small gloves on as they just "stick" to my skin and won't slide on. I do have dry hands and use O'Keefs working hands lotion. Could this be interacting with the hand sanitizer? Although I don't always have lotion on my hands. Any ideas? In my case would you just use the size medium? I keep getting told that I will have trouble finding the vein without gloves that fit tightly to the skin. (I also want to mention that size small fits tightly, but doesn't extend to the wrist like size medium, bit the medium doesn't fit snugly without any "extra wrinkles" in the glove. I did use size small while in school, but we were able to eash hands every time rather than using hand sanitizer.) I am at my wits end here. Thanks everyone for any ideas.