I may have advice after a few questions, but first....
Hugs!!! You are NOT a failure. You can do this.
They are 4 and 6? Do you live in a strict state or lose? Can you teach them together for say history, science, art, and music? My boys do science altogether (3,5,&7).
My 7 yo CANNOT be quiet. So we do absolutely everything in question format. He can pretty much stay quiet long enough for me to ask him what he knows about a subject. When he has finally talked himself out of the topic, I have him read about it aloud to me and younger brothers. Let's take language arts. I'll ask him what's a noun and he'll go on for a while about it'd a person place or thing and gives me examples. Then I'll have him read me a poem (in a sleeve) and circle each of the nouns he sees. Sort them into person place or thing. Then I'll find an idea and ask him if it's a noun. He says yeah, but can't figure out what group to put it into. While he's thinking, I'll have the five year old name a couple nouns or rhyme with the last word we found. When Zave can't hold back his tongue anymore he'll say he gives up and we'll talk about ideas. He goes off to make a page in his language arts binder about nouns. While the 3 and five yos help me find the school house rock about nouns. We watch it until they can just about sing it (just cause they like it). Then we grab something new.
Also have you tried any books on tape with your youngest? My 3 yo will sit with ear phones on and listen to one book which buys me about 15 minutes of quiet time with the oldest.
Remember that it will just keep getting easier in the way of attention span and tantrums.
Usually a tantrum is their way of explaining something to us that they have no words for.
Hi dear!
My first advice is to check out the laws in GA. Make sure you are in compliance with all those for your state first and foremost! 
My next piece of advice....at this age, is there a reason why you feel the need for them to be quiet to learn? I don't mean to sound rude and I hope that I'm not coming across that way....I only mean to be supportive and full of hope and advice. 
There are so many activities you can do where the kids can be loud and active and still learn. Check out your local parks and see if they have nature trails. Take the kids on walks there and see how many of the local wildlife species you can identify (loudly! haha!!). Make it a game. If you see a lizard....shout it out! If it's a bird...whisper.... If it's a squirrel...SHOUT!!! If it's a chipmunk....whisper.... You get the idea. If someone makes the wrong sound, that person has to make a monkey noise or a cow noise or something of the sort.
You can take them outside...or even set it up in a playroom or living room...and learn letters (or words for your older child). They have their back to the room while you scatter papers with the words (or letters) on them around the room. Then, you step back and call out a word. The children turn around quickly and race to find the word. You can give prizes for the child who finds the most words.
There are SO many activities you can do with the children that facilitate learning but don't require them to be quiet. Perhaps you just have active learners. There's nothing in the world wrong with that! My son loves to lie on the couch with his feet/legs where your back should be and his back/head where your legs/bottom should be during reading time (I have a pic of that somewhere but can't put my hands on it right now.) My oldest daughter....well....this pic speaks for itself.....
She doesn't do all her reading while hula-hooping but, she was tired of sitting still and needed some movement.
Good luck in finding a method that works for your little ones!! And....have fun in the process!!! 
Sometimes I think of kids like sharks. They can sent blood and fear from a mile away. If you give up, or scream, or give in to their screaming, then they see screaming as an effective tool for getting your compliance. It's like Pavlovs dog, if you ring a bell everytime you feed a dog, then eventually if you just ring the bell the dog will salivate. If they scream, and get their way, then they will always scream in order to get their way.
I couldn't care. Less if my youngest flailed while in time out, as long as he didn't leave his thinking corne. If he left it, then he would be put back and the timer would be restarted. I think the longest he ever spent in time out was 2 hours, he was testing my mettle to see if I would follow through.. I don't spank, and I don't scream, though I have my drill sargents voice that is very firm.
But here's the thing. I've learned so much from my special needs child, about how to effectively communicate with my boys and good expectations. I can not expect a 5 y/o to act 25 and sit quietly for a lecture. No matter how interesting the material, it is just against their nature to do that. So I can't then be frustrated that they don't meet that unrealistic expectation.
Now, I have an anxiety issue as well. So I know how it can mess with your mind. But sit quietly and really analyze what you hope your kids get out of schooling. NOT what you hope the world outside your family thinks (like meeting state standards or whatever), just really think it. On the day after the kids graduation, what are your top things you hope they have taken away from the experience?
Here are my things, ese work FOR US.
Behavior Bucks - http://www.cafemom.com/group/114079/forums/read/16866109/Behavior_Bucks
This post defines exactly what our HS is like - http://www.cafemom.com/group/114079/forums/read/16483391/Defining_Our_Homeschool_piog
Hang in there!
Home Educators Toolbox / Articles / Kicbuttmama's Crazy Lapbooks / Kickbuttmama's Home Education
Albert Einstein --
"Everybody is a Genius. But if you judge a fish by its ability to climb a tree, it will spend its whole life believing it is stupid."
I live in GA and it is a fairly "loose" state so far. All I have to do is provide the equivalent of 180, 4 1/2 hour days of instruction for my child in the areas of language arts, math, science, social studies, reading, wrting, and some other thing. I have to submit a monthly attendance form at the end of each month and that is all. At end of her third grade year, she is required to take the CRCT test but I keep those results, not the school.
As for teaching the kids together, I find (I have a 5 year old and 2 year old) that it works to my advantage to teach together where possible. As for discipline, they are just now entering the "fighting" stage. I used to be an advocate of spanking and will spank if necessary but have found a technique that works for me - the calm down corner as seen on LifeTime's America's Super Nanny. It is not for the faint of heart. No matter how long or loud the child screams, they stay put until they can be quiet for three minutes (3 min regardless of age).As the parent, you cannot back down or give in. It is a power play between you and the child. My 2 year old was in it for an hour the first time we used it. Now, all I have to say, is "this is your warning" and he immediately (sometimes) complies. You only give ONE warning and then you follow through without fail. Consistency is the secret ingredient in any discipline. I used this in public at the checkout lane and it worked! The calm down corner was in front of the soft drinks. My 5 year old was running around and I gave her a warning. She thought I couldn't do anything, but I showed her. I put her in front of the cabinet and set my timer on my phone. She got the message. You have to be consistent which is easier said than done at times, I know.
Kids know when we are tired and stressed and always seem to choose that time to want to get their way. Stand strong and you will see a huge difference in no time at all.
You are NOT a failure. You are a mom. <<<<<<HUGS>>>>>>>>
NEED HELP WITH HOMESCHOOLING INFO. WHERE I LIVE IS NOT HOMESCHOOLING FRIENDLY.
- sourflower
on Jul. 12, 2012 at 5:08 PM