Your grip is definitely NOT stupid. I've been straight shaving for over a year and had always had to turn my head in a funny angle to try to see if I was starting the stroke on my sideburns at the right spot because my hand was in the way. When I visited Bob Allman (a custom razor maker in Houston, TX) he showed me that he shaves with the same grip you're using. It allows you to see exactly where you're starting your downstroke. I've been shaving with that grip for the WTG pass on my cheeks and sideburns ever since. The added benefit for me is it also allows me to define a crisp edge on the lateral margins of my VanDyke as it's easier to watch the toe of the blade vs. the heel in this area.
I use microfiber wipes to clean the blade and use a stropping motion. Many will tell you to know rinse for fear of dinging the blade on the faucet. I'll have to admit, I sometimes rinse under the faucet, but I'm extremely slow and deliberate with my movements. I bought a package of microfibers for very cheap at WalMart (I assume you have a similar store down under).
Could you tell us a little more about the drying lather? Is it drying in the bowl, on your entire face, or is it drying out on sections of your face before you get to them?
Greg