I really liked what both Laurie A and Isn'tThisFun suggested. They made some very good points.
We chose not to put Kiddo into an extracurricular this season because it was his first go-round doing full-day school. (He attended half-day kindergarten last year.) I am very glad that we chose this route. School gets out at three, we walk home, may linger at the playground or walk with a friend, home around 3:30ish. I have a checklist for my six year old son that he is able to follow independently, so I can just keep referring him "oh, go see what's next on your list" and he's been remembering more and more of those simple tasks (empty backpack, give folder to mom, unpack lunchbox. etc.)
By the time we are done with a snack and his small homework, it's usually about 4:30. The afternoon flies by for us. He usually has time to help set the table and do something he enjoys for about an hour before we're rolling into dinner. It makes me sort of sad that he only has one hour out of a day to do what pleases him, nurtures him.
I am a person who thrives on routine, so our days feel relatively predictable, which helps him.
If it were me, I'd pare down on the extracurricular activities. Let them each choose ONE that they really, really want to attend. Being under instruction all day is something that some kids might thrive on, but your kids aren't expressing that.
I like the idea that Isn'tThisFun suggested of no talking for a while. Sometimes, when my son gets on a whiny jag, I just tell him "I think it's time to be quiet for a bit. Maybe you want to think about how to (solve that problem/what you could do next time)." and even without siblings, six is a notorious age for being ungrateful, selfish and wanting 'more' and answering back. Yes, it is horribly unpleasant and we are also dealing with this as it comes. It's disappointing and often very ugly sometimes, but I am firm about "you may stay in your room until I call you out. Why? Because you are so unpleasant to be around, you need to go take a break!" (It's a sanity saver!)
Also, be sure the kids are getting enough sleep and down time. We do a seven o'clock story time (pajamas on, teeth brushed beforehand), followed by a 'stay up privilege' of 30-60 minutes of quiet play or reading, depending on if he has school the next day. We forget, once they start school, to make sure they have some quiet time each day to play alone. This is time my son will use to read or draw or play with his Legos.... he needs this to wind down. His actual bedtime is 8 on weeknights and 8:30 other days. He usually sleeps until 7 if you let him. Sleep is so important.
I do get where you are coming from and how draining it is. Hang in there. Use the advice here which works for you. Good luck!