Handshakes and all other activities involving hands can become awkward when dealing with sweaty hands. It is something that is difficult to control, and often gets worse in stressful situations. Aka., the sweating starts when we least want it.
Washing hands regularly, just water is fine, takes away the sweat and cools off your hands. Having a tissue in your pockets or in your bag is an excellent idea, as you will have easy access to dry your hands regularly and quickly. Try to observe when the sweating gets worse, and see if there is anything you can do to solve the problem. For example; if you sweat more during business meetings, do something before or during the business meeting that calms you down.