Glad you're here, Kimberly!
I survive homeschooling two (not three! Yikes!) active kids because of two little words: CHILD CARE!
Just a few hours a week makes all the difference to my sanity. I didn't have much/any (that's child care, not sanity, though the sanity soon trickled away, too!) at the start of the year. It was so hard. One particular day, I was on the verge of verbal abuse, I think. Afterward I had a long talk with the kids and apologized. They forgave me, and I've tried to forgive myself. On the bright side, it became clear that something had to change so that I NEVER went to that bad place again!
A few hours of child care each week made it all better.

A sane mommy is a happy mommy! So that's priority #1. Find a way to fit some sort of time away into your schedule and your budget.
Also, I encourage lots and lots of independence out of my kids. If the almost-2yo wants schoolwork, give her some! But make it something you don't have to hover over. The more they can do alone, the better. It sounds like it's time for the 6yo to start looking up things for himself online or in reference books. Can the 6yo teach something to the other two kids? Use what you've got!
Finally, art supplies (crayons and clay) and manipulatives like tangrams and Legos and Lincoln Logs are your friends, just in case you haven't discovered how useful they are as child-distractors yet. One or two kids can be assigned to build a dinosaur or write a book while the other 1 or 2 are doing something with you. I even use the TV if I must for a half-hour here or there, though I'm not at all big on TV in general for my kids. But TV works wonders for keeping one out of your hair so you can work with another. Tune in to "How It's Made" or "Sesame Street," and it's not such a bad thing, even.
Just be sure to give yourself time away somehow. Quiet time for everyone in the afternoon--naps not required--or a requisite "out in the yard without bothering Mommy" time is a good, good plan. As they say on the airplane, put the oxygen mask on yourself before you attempt to assist someone else. That goes double for moms! If you go insane, everyone else will be close behind you, too. Make time for you!
Does that help at all? Or is this all obvious stuff I'm telling you? Do you have any thoughts to share? I'd love to hear what you're doing to stay sane, too!