I didn't do it as a single parent, but I've sailed with a 4 year old and 21 month old before
My best advice would be to keep your expectations reasonable about what you'll be able to see and do. Be ready to recognize when they've had enough and need some sleep. We picked the 2nd dinner seating because we were afraid our kids might be disruptive and so we wanted to keep the damage to a minimum...but that was a mistake! Two nights out of three they were so tired that they fell asleep in their chairs at dinner. If you somehow get stuck with 2nd seating, you might consider just skipping the MDR's and eating where/when you want. If your little ones are like mine were, the idea of sitting patiently through a multi-course meal is a non-starter, and that means you likely wouldn't enjoy your meal either.
I wouldn't advise trying to book a whole bunch of excursions or anything like that. I'm a big time planner, and so it was hard for me to just "go with the flow"...but in the end that was the best strategy for us.
We used the nursery for our smallest one evening when we dined at Palo, and we were happy with the service they provided. It's well worth the cost, and so I'd advise you to take advantage of it and the kids clubs to get some "me" time (like a spa appointment or just lay out at the pool and get a nap). If you want to meet other adults, there will be singles activities in the navigator.
Bottom line...there will be plenty to do on the ship, just let the kids' energy level be your guide on what you do. And make sure you make some time for yourself, the kids will be just fine without you for a little while!