i don't know about verizon, but i read that sprint phones ship unlocked... and if it is locked, i would just have to tell them that i'm going to a part of the world that lacks sprint gsm coverage, and then they'll give me the code to unlock it.
a quick google search lead me to
this which on the second page basically says what i just said about sprint. just call and say that you're going to a part of the world without verizon coverage. i'd tell you where exactly, but i'm too lazy to look at the international coverage maps, but i'm sure you can find a spot.
and yes, unlocking it will still allow your phone to function. when you unlock it, you're basically just allowing the sim slot to work with att and whatnot.