I use Accuvue 1-day Moist contact lenses. I wear them for a day then throw them away. And I always bring a couple of spares in case I need to replace them. The nice thing about disposable lenses is that you don't have to worry as much about contaminating them, since you only touch them when inserting and removing them.

As SBRDude suggests, you don't necessarily have to use "clean" water for the entire process of getting your hands clean. Here's what I'd suggest:

1) Wipe the majority of the dirt and gunk off your hands before you begin washing. You can use your shirt, pants, a rag, towel, or anything else you happen to have handy.

2) Use pre-packaged, cleaning cloths or baby wipes for the majority of cleaning. My favorite product is Dymon-brand Scrubs Hand Cleaners (not Hand Sanitizers). I buy them by the bucket and use them to clean my hands after working on cars/bikes/motorcycles. The Scrubs cloth is a bit rough and contains an embedded hand cleaner; one sheet is usually all you need to remove all of the dirt/grease from your hands.

3) Wash only the tips of the fingers that will touch your contacts with clean, potable water or saline solution. This is another advantage of disposable lenses: you don't have to scrub them around in your palm to clean them, so you really only need the tips of your fingers to be clean.