Thank you all so much for the advice! It's so good to know that I'm not the only one hanging by a thread, lol.
We're involved with a homeschool co-op, but we pulled DS6 from school so abruptly that by the time we joined the co-op was winding down for the year. Hopefully when the classes start back up it will help some. The kids will all have 2 hours of classes every Monday (I'll be in the room with DS4 and DD since they'll be in the same group and she's so young).
We try to get out of the house a couple of times a week, but with gas prices like they are I end up combining several errands in one trip which is too much for the kids to handle. Summers are hard in general too because we live in the South and between the heat, the humidity and the mosquitoes it's completely unbearable outside.
I think I need to be more vocal about me needing to be away from the house. My time away is usually to grocery shop, lol. We can't afford to hire anyone other than the occasional babysitter for dates, but DH is perfectly capable of caring for the kids. I've been considering joining a book club just to have something to do that's semi-intellectual and completely unrelated to children.
DS6 is becoming much more responsible for his school work. He does his language arts and math on his own, history and science we do together. I give him a folder each week with a to-do list and worksheets paper clipped together for each day (the sheets are just grammar and handwriting). The list includes what to read in his math book and the problems to work on. Except for when we're first working on a concept, I'm just close by to help when he gets stuck. I've found that he fights much less with the notebook telling him what needs to be done than he does with me. It also lists his chores for the day and any special activities we have. The only rule is that his work has to be done before noon or he loses TV for the day, simply because I've found otherwise he will drag his feet until bedtime.
I am trying to teach him to look up the information that he wants online or in our encyclopedias. So far the thing that works the best is for me to walk him through what I'm going to look up and then let him read when I find the answer.
The big frustration is when DS4 is doing his school work or if I'm trying to read the history book with DS6 or I'm doing science with the boys, DD feels left out. She sees that everyone but her is involved in productive activity and she knows the difference between busy work and actual work (she's like that in everything, I can't give her dish towels to fold, she wants to fold shirts like I am, I can't give her silverware to put away, she rolls her eyes and starts handing me plates from the dishwasher). I try to involve her with the schoolwork when I can, but the fact of the matter is that she's too little to actively participate and is mostly in the way, distracting the older 2.
I'm just really not sure what I can give her that will keep her occupied, be productive and not require more than general guidance. Neither of the boys was interested in actual work at this age, they were much more passive about learning. And leaving her to her own devices is horribly messy, I left her to play while the boys and I were working (our house is small and we were downstairs with her), she defeated the lock on the refrigerator and covered the entire kitchen floor with flour, she was so quiet about it I had no idea until I realized she was being so quiet. She isn't doing it to misbehave, but a bored DD is a destructive DD. It never would have occurred to the boys to do some of the things she does and she regularly catches me off guard.
I know there's no magic cure, a lot of this is just going to take some time for them to grow a bit. It really is comforting that I'm not just a wimp, lol.