Hi there Super Mom!
As a starting point, you should look at my posts about strollers found
HERE,
HERE,
HERE and
HERE. The difference in our situations is that I was only dealing with three preschoolers, whereas you have four, so I'm not 100% versed in the circumstances you'll be facing.
The two-doubles that you mention would certainly work, and may indeed be your best option. My only possible concern is that I wonder what would happen if you wanted to take just your five-year-old out to do a couple of bigger-kid rides while the younger ones nap. Of course you could always just have your oldest son walk in this situation. But if you wanted to take him out in a stroller, then it would be somewhat cumbersome to have just him in a double. Possibly a double for the twins and a sit n' stand stroller for the older kids might be more manageable, if the five-year-old is willing to stand, and if you have ready access to one.
Your choice of the Contemporary as your resort was a good one. The monorail access means that you'll have an easy time getting you and your strollers to both the Magic Kingdom and Epcot. Please be aware that getting to both the Animal Kingdom and Disney's Hollywood Studios involves taking a bus on which all strollers must be folded. That may also play a factor in your stroller decision making process.
Good luck with your planning and enjoy your Walt Disney World vacation.