To wash hands from tum'a or all other purposes, you do not need to dry your hands first--except before washing for bread, and then ONLY if the person who washed hands before you:

• Did not use a revi'it of water (per hand) to wash his/her hands, AND
• Only poured once on each hand.

In sum, you almost never need to dry your hands before washing them!

However, if the person before you had tum'a on his or her hands and poured only once, the tum'a will be transferred from his/her hands to the cup.