Hi jessicad312, and welcome to cruise-chat.
No, you are not crazy. Cruising is a wonderful family vacation. We've been cruising as a family since 1994, when our son was nearly a year old.
As you have extended family coming along, they may be able to offer you one night of "couple time" by watching your youngest child. The 3-year-old will be old enough for the onboard kids program.
Cruising with kids is actually easier than other vacations with kids. Your meals, transportation, and entertainment are all taken care of once on board, and all of you will be pampered by the crew.
As for excursions, I see you are going to Nassau, Great Stirrup Cay, and King's Wharf. It's been a while since I've been to Nassau, so I'll leave that to someone more capable of answering. From King's Wharf, you can take a ferry into Hamilton or St. George -- IMO, St. George will be easier with kids, having done both. You can also take a bus to one of the South Shore beaches. Great Stirrup Cay is a beach day on NCL's private island.
I recommend the public transportation over taxis simply due to the car seat issue. I wouldn't get into a taxi or van without one, but the ferries are perfectly safe. You can roll the stroller right on.
If you plan on doing ship-sponsored excursions, I'd recommend something that is short and does not require you to spend many hours at a time on a tour bus, as the young ones might get bored. I'd also look into the form of transportation being used -- again, some will use cars or vans.
Any other questions, please ask!