We're kind of in the same boat - I had five in three years and two months. So I understand about being overwhelmed at times. But I believe that God (or your version of your Higher Power) does not give you more than you can handle.
That being said, it's still hard!
I loved when my girls were little tiny (6 weeks) and would reach for each other, as if they already knew they were not alone and needed each other. I loved as they got older that they had a buddy. I think one would be much harder around 18 months or 2 years because a singleton constantly wants your attention, but between your twins and older kids, they'll always have someone to keep them busy.
It's nice having two for the older kids to "help" with so they are not fighting over one baby. And I have to agree with the first time they looked at each other and smiled while we were tandem nursing being right at the top of the list. And when one is sick, it's nice to have a comparison with the other twin so if they're identical, you know what the sick one "should" look like. Mine got RSV at 2 months (I took them out in public WAY too much because I was just plain uneducated) and I knew one was definitely sick because she was more blue than her sister.
The understanding that my girls now have (at 6) that they were in my tummy together is really precious too. I loved when they fell asleep together (and still do) in a pretzel-style pattern. It's really fun when they're little to see two little heads right next to each other as they nuzzle their noses together.
You have a lot to look forward to and you're going to make it through this. If I can do it, anyone can! I don't think your homeschooling plans need to go out the window either. You can teach your children a lot about breastfeeding, diapering, family dynamics, and baby care in general while the babies are young. Before you know it, you'll be on a good schedule and you can plan in school time more easily. But for the first little while, just focus on educating them in ways the public school system can't, about babies.
I put my girls in kindergarten this year because I am in school, but we are going to homeschool next year (I asked the girls and they still would prefer to be at home) even though we are having another baby (or two!)
You go girl!