I roll mine out with a rolling pin, and drape it over the pan, making it large enough that it hangs well over the edges. Then I place the cheese and toppings it, building up.. once you're at this point, the toppings, not stickiness are holding the dough in place. when everything is in, I cut away the excess dough and bake. great results every time. nice, very tight fit in the pan.. perfectly uniform shape and cooking.
A few procedural things to look at. Deep dish crust needs to be crisp, yes, but it's the flaky texture and richness that really make it unique. In many ways, it's more like making a biscuit or pie dough than a traditional leavened bread dough. This being the goal, the oil and water additions are backwards in your procedure.
Try starting with a bowl of flour mixed with the sugar and salt. You NEED salt. Trust me, I'm a chef.. a small addition of salt, say just a teaspoon of kosher salt, will greatly enhance the flavor of your dough, without making it salty at all. Store-bought bread for example, would taste lousy without salt... it's gotta be in there. Now add the oil and mix until it's well incorporated, forming fine, flaky bits. Then add your water and yeast, mix, and kneed for a few minutes. Form it into a ball, and cover it tightly. Let it rise at room temp for a few hours until it's doubled in size. Punch it down, cover, and let it double again. Now pull it out of the bowl, punch it down, and roll it out. The difference in texture is great. As far as greasing the pan, a light coat of butter will do just fine.. wouldn't want any more than that. there's enough fat in the dough already to get that great texture and flavor. Great stuff. good luck.