Never leave home without a tube of your favorite hand cream in your bag, and use it frequently throughout your trip. Not only will this help keep your hands from drying out but can help counter the effects of harsh commercial-grade soaps found in public restrooms.
I am currently using a tiny very-TSA-friendly bottle of Bloom Rosemary Infusion Hand and Body Moisturizer from the Embassy Suites Hotel. It feels good and smells great. When the lotion is gone, the bottle is perfect size for refilling with my favorite hand creams from home.