In response to what you think is NOT shouting (not your original post; your follow-up post), check out Netiquette Links (pick anyone). You know, just think what it's like to have someone you DON'T know shouting at you, especially when they want help. Really, and I mean really, between you and me, it's a real turn-off BUT no big deal. (Like me, you are still learning every day, right?) I'm sticking around because I can take it every now and then.
Moving right along, I don't think a weed sprayer is going to have the pressure to do the job. If it does, you'll be spending a lot of time doing it. As for a cleaner, if it's dust you're trying to remove, there isn't one. If you're trying to remove crayon marks or paint or something like that, well, then you'll need a cleaner. Use the wet-vac to first vacume off the wall. Then remove the filter so you don't ruin it when you suck up any water. Have someone 'follow' you around as you wash. Keep an eye on the water level in the vac. It IS heavy the more you fill it. If you want to throw juuuust a little mild dish detergent into your sprayer or a spray bottle attached to a hose, then go for it! Hey, I just though of something ... Maybe using a hose-attachement spray bottle where you can regulate the water volume is the answer. I know they're in the home centers and nurserys. Low water and high pressure is what you're looking for in the bottle attachement.