From what I know, Disney in the summer is sweltering and the lines are so very long, just when everyone is traveling because it's cheaper. We've done Disney a few times, and while it's terribly overpriced, it's a unique experience. I do think you have to be reasonable - there is no way that one day at Disney World will even begin to satisfy the kids' feelings that at least they have been there! See if you can squeeze out another day or two, and put the kids on a budget (I think you're good with that stuff!), limiting them to a fixed dollar amount for souvenirs and also making them earn some of their spending money (from sources other than you). Even an 8 year old can walk a neighbor's dog or save his money from Grandma or forego a big birthday party to bank the money.
You can keep your costs down by getting an apartment vs. a hotel, and making your own breakfasts and taking your own lunches and snacks - Disney definitely permits outside food, and there are plenty of places to eat your own fixings. Everyone can pitch in on dinners and making breakfast, washing dishes, etc. so it's not all on Mom! You can splurge on a nice dinner at one of the parks at whatever theme area interests you. Taking the family to the inlaws' is great too and saves a lot of money.
I feel there is value in family time before those teens move on to college or jobs and have a totally different schedule. I know that pulling kids out of school is a huge problem in many districts. But we always did it. If you do 2 days instead of a week, it's no more of a problem than when they're out sick. And yes, they might miss something, but if you're trying to create memories, it's okay for them to learn that life is about choices and a few sacrifices. Surely you can avoid things that would get them bounced off a team and maybe miss something less critical. Try to figure out what will matter the most in 5 years - photos of together time, or a major sports opportunity. Only you can answer that.
But I live in Massachusetts like you, and I'd suggest the April vacation next year for the big trip. No one else has the same week off 90% of the time (check for Easter's timing) because of Patriot's Day which is only observed in Mass. and RI. Take the kids out on just the Friday before the vacation - no big deal. Florida will be warm and lovely, but not a humidity house. The lines at the park will be much more manageable. My neighbors went in July and after a few days, the little kids were completely fried. Be willing to split into 2 groups to take the younger kids on some rides that don't interest the big kids, and vice versa, then meet up afterwards. That often makes things pretty appealing to the older kids who want some independence now and then and may want to go on things that are too scary or have too high a height requirement for the 8 year old. Sometimes the 8 year olds just want to hang in the pool and the 16s are go-go-go!
It sounds like you are very organized with the finances - you save from one job, you can tutor, you separate your emergency fund from other funds, and you have looked into budget plans for braces and so on.
Also look into places where you don't need a real passport. For example, you need good ID but the kids don't need passports for many countries (check with the state department for specifics), and places like the Virgin Islands and Puerto Rico are US territories. Puerto Rico, for example, is really fun because it has so much variety - el Yunque rain forest, Phosphorescent Bay, historical areas of Old San Juan (forts and old cannons and Spanish colonial history), and more. That might be even more fun than Disney - something for everyone.
Your husband's issues with money have nothing to do with the trip - it sounds like he's feeling badly that he's not the breadwinner.
So maybe it makes sense to do something cheaper this summer so you still get away but save the money for the spring "big trip". Instead of Cape Cod, try something less touristy. Or go for a lesser home with the kids in bunks instead of their own rooms. You're on the beach or on the deck half the time anyway. Play more board games or do less stuff with ticket prices. Economize anywhere you can, but keep socking away those small savings into a separate account. Give each kid a "vacation" jar and teach them to put in everything from the bottle return nickels to a few extra bucks they earn by helping out a neighbor. If they go out on those snow days and shovel a neighbor's walk rather than sitting in front of the TV where it's nice and cozy, the cash adds up. Those are good lessons for all kids. If they don't have savings accounts, open some - there are no-fee accounts for kids under 18. Then the oldest 2 can morph into student checking accounts for when they leave home.
And definitely sock some money away where your husband isn't aware of it, and then surprise him with the total! Yes, I know we all have to save, but really, sometimes we miss the joy of the moment by saving for the eventuality.