The ride won't be affected, even with the leveling kit, so much that you won't like it. a 33" tire in a mud & snow or mud terrain tread pattern will be a lot noisier than the stock tires. Handling may feel a bit weird at first as the tires are wider and have larges spaces between the tread blocks, which in a slow turn situation like in a parking lot, it may feel like they squirm a bit.
In your case, as is the case with many others, you may not need that aggressive of a tread if most of your miles are on-road and most of the off-road use is mild. In that case, an All Terrain (A/T) style tire may suit you better. They still give good off-road traction when needed.
As far as size, a stock 20" wheel is 8.5 to 9.0 inches wide I believe and I'm leaning towards 9" as stock. In that case, a P275/60R20 is right at 33" tall and fits a 9" wide wheel just fine.
Other size choices in LT (light truck) tires that fit your rims would be:
LT275/65R20 - Just a shade over 34" tall and will fit with the leveling kit installed.
LT285/55R20 - Just a shade over 32" tall.
There are other sizes available to choose from depending on what brand you go for.
There are quite a few pics of trucks here with stock 20's with 33's and a leveling kit on them. It's a pretty common upgrade.