We just did the same cruise last Friday. I got to the port at 9:30 AM and saw other people disembarking still from the previous cruise. They let us in the garage with NO issues at all. Told me to park at level 3 (you pay for parking when you go into the lot, not when you leave after your cruise). We got in line with our luggage on level 3 in the garage, with about 5 or so other families in front of us. A porter came through collecting luggage from the line, and we tipped him accordingly. They opened the security point at about 10am, then we all moved down the bridge to wait to go into the terminal. We went through security inside the terminal about 10:15ish. We went right into the check in line with our documents ready (cruise contract and ID's), got our pictures taken, given our Navigator (printed schedule of activities), then walked away from the line and were planning on heading to the Character greeting line (they weren't out yet) and were chosen as family of the day, so we skipped the lines and we boarded first at 11:30. By that point the terminal was jam packed though. So my motto is, the earlier the better!
P.S., we were not asked for ID's when pulling into the terminal lot before getting to the garage. The only point we were asked for ID was to get through the first checkpoint before the bridge. There is no more "preferred" parking, it's all the same now, and the price is $22 a day.