Your process sounds reasonable to me, especially as stateroom type isn't important. Worst case, you find yourself with a week or more in Seattle and the surrounding areas, which isn't that bad.
I looked at Royal Caribbean's site, where they have an engine that helps you determine the earliest time for your departing flight. I entered Rhapsody of the Seas, disembarking June 22, and the recommendation was 11:45 am or later. It shouldn't be much different for other lines; we're taking HAL, and I recall something similar.