This is a pretty big flaw.

I would highly recommend the developer add a simple voice recognition functionality. Even something as simple as "next" & "back" which scrolls the screen would improve the usefulness of this app.

Use your pinky to interact with the phone. (Seriously, I keep my iPhone in the kitchen when cooking and my pinky generally doesn't get greasy.)

Same way as in the bathtub. Zip-loc bag (or competitor).