That's funny, my hands sweat thinking about and watching climbing too! I don't like heights but love climbing. Once you learn to trust the gear it's not scary in a gym. Outdoors it's a different story because of all the unknowns but manageable based on the route you try. You should give indoors a shot.
I think it's funny too. Watching videos make me sweat like crazy but actually climbing doesn't do it as much. It makes climbing tougher but you can chalk up and it helps a lot.