Wow, you really don't have glasses? I can't imagine. I've been backpacking with contacts many times. I've never had much of an issue with getting my hands clean enough to take them out and put them in--my problem is doing it without a mirror, or with only a small signal mirror sitting on a rock or something. I always bring glasses for wearing after I stop for the day and take out my contacts.

It just sounds like you're a little too OCD about getting your hands really clean, or you're trying to carry all the water you need for your trip rather than refilling and purifying. If you bring CampSuds or some other similar soap you can get your hands clean enough in whatever water is around you (stream, lake). You might also try taking that waterless hand sanitizer. Just make sure you wipe the excess off and wait a while for your hands to dry. Use iodine or chlorine tablets to purify water from the wilderness, or get a filter like the Platypus CleanStream or something. Or just carry more water and drink less. For a two day, two night weekend backpacking trip four liters of water might be all you need if it's not super hot and humid, and if you can cook with whatever water is around you.