When we travel as a couple or bring our college-age son, we always prefer the second seating for dinner. We've never had a problem getting a private table for the two or three of us, depending on the cruise. We have the perception (and probably a lot of truth) that you will get fewer families with small kids as they generally prefer the earlier sitting so they can pack off the little ones to bed. That doesn't mean small kids aren't there...just fewer of them.
It never hurts to make your requests. Ask for your private tables, preferred dining rotations, and even a preferred waiting staff. The worst that they can say is "no" and you're still going to get great service and great food. Ask for them ahead of time, and then ask again when you get to the port and check in.