Yes, deep breath. First, congratulations! You can certainly do this for what you've saved, and probably less, if you want, providing the whole cost of the cruise is covered. As was mentioned, $6k may actually not cover the cost of the cruise in what the cost is today. Assuming the cruise is covered, and your parents will take care of airline tickets...
You can spend next to nothing once on the cruise. The nice thing about
DCL is once onboard, everything you need is already included. You do not have to do any DCL port excursions. You can DIY pretty much everything in port, just walking and enjoying the towns. If you decide there is a paid activity you'd like to do, it's usually very easy to arrange it yourself for significantly less cost and more flexibility than DCL excursion. Ask for help how to do that here. You could literally eat every meal on the ship, and pay nothing for extra food for the entire duration of the cruise. Activities on the cruise are "free." You can find a list of what costs extra here - it's not actually very many things (alcohol, popcorn at the movies (?? never understood that, since the movie is free), the spa, shopping in the store on board, pictures that the ship photographer takes, etc), all of which are not necessary for a good vacation. Tips are usually given to certain people at the end of the trip - cabin steward and servers. That may not be included in your prize.
You'll find good tips here on getting to and from the cruise. I definitely recommend flying in a day early, to decrease risk of missing the ship due to unforeseen circumstances. There are minivan taxis available everywhere I've ever been that would fit your family and luggage just fine. You can find plenty of reasonably priced hotels. You can even use hotwire.com and sites like that to find good deals.
You could probably borrow luggage from friends. It's one of those items that mostly just sits around in someone's closet anyways. You've already got the right clothes for Alaska, coming from Colorado.
Phone cameras can take some pretty good pics, but if yours doesn't, borrow one.
The passports are a necessary cost, but will be good for 10 years for the adults, and 5 for your kids. Who knows, maybe you'll use them again?
So, definitely not a nightmare. Just surprising and overwhelming when you weren't planning for it! Who ever expects to actually win something like that??