Kudos to you for admitting this. Not to trying to be unhelpful at all in being harsh, but here is what I think of this COMMON scenario.
The way you need to not yell is to UNDERSTAND that you are raising the chaos levels by doing that. You think you are yelling because they are making you, but really, your ineffective yelling is just swirling around in the mix with their bad behavior. And you are forcing you kids to build up a tolerance to it. They will grow to shield themselves from your shrill annoyance through many tactics, and they will like you less as a person over time than they would if you respected them and were not a frantic yeller. My brother and I never minded, and connot remember, the rare firm discipline we got as kids, but the few bad memories we have are of the few times my parents yelled at us angrily. I can't imagine living with that daily. To this day I cringe when I see people yelling. My husband and I yell at each other on occasion, and it sucks, and we both hate it, but we're human. We always apologize so the kids see we know it's not nice, and we rarely do it.
You say you've tried all these other discpline techniques, so now you yell because it gets their attention in the moment. However, since you are always doing it, that is PROOF it works no better than any of your other tactics, since ideal discpline teaches correct behavior and stops repeat offenses.
I have a friend with this level of chaos at her house and it sounds EXACTLY like what you are describing. She even has the kids on meds for sleep.
Believe me when I tell you your other tactics have not worked because you have not been CALMLY CONSISTENT with a FIRM ENOUGH one EVERY TIME the behavior STARTS.
If you are yelling, it is because several warnings have not been heeded, things have escalated, the kids are nuts, they're doing all the same old stuff, and you're PO'ed, right? You never just wake up and start yelling the first time someone does one tiny little thing do you? You need to take control of how it reaches that level.
What has worked in our house-where we literally never yell at the kids (they're not teenagers yet :) unless someone is far away and can't hear us, and the ONE time I yelled my kids almost fell over from shock, is centering yourself to be calm and direct at the beginning of a wrong action. Give one clear warning what you expect and what will happen if it is not followed. Enforce immediately if the wrong action is taken. This may take lots of repetition since the kids are used to waiting until you're yelling to "hear you", but it will not take any longer than continuing to yell, which may NEVER work.
Be sure you are zeroing on real behavior issues that need to stop, not just things that are annoying you. For instance, enforce tantrums, talking back, aggression, but don't get mad and scream because someone is being noisy, etc.
Also, if you implemented discipline, but then it only worked for 5 minutes, you don't then graduate to pointless yelling, you repeat. Calmly.
Today my 3 year old son was disciplined. He rarely is. He is an excellent, sweet 3 year old boy. He is happy and affectionate and well behaved and we are VERY close. He has been getting in the habit of making a funny face when his father or I are reprimanding him rather than saying , "yes sir" or acknowledging what we are saying. It's a typical respect glitch most kids go through. It's not a big deal, but we dont' want it to blossom into nightmare disrespect. We have talked it through with him several times when he does it without discipline to be sure he totally understands the correct way to take a direction without doing blinky eyes and shruggy shoulders etc. Today my husband was saying something to him, and he snapped, "I KNOW" in a bratty tone. He knows this is not allowed. His dad then reminded him CALMLY not to talk back, and rather than stopping, or saying "yes sir" he started doing the eye blinking shoulder shrugging thing and said "I KNOW" sarcastically again. So his dad walked over and popped him on the butt. Not very hard, but it was a reminder that he was WAY out of line for an offense he was absolutely clear on, but this was his first enforcement of it. I'll be shocked if he does it again. That is because he is used to absolute consistency and knows he will never get away with it if he does continue to do it. He cried like no tomorrow and then apologized on his own and got a hug from dad, at which time my husband explained in a kind tone of voice AGAIN why he needs to be polite when we are telling him something, same for teachers, etc. All the while, no one yelled, and no one got mad.
You may be thinking, "yeah, but he didn't do anything, it's not like he was hysterical and ripping up the house". But the key is that he would never be allowed to get that far. Calm warnings suffice 99% of the time. Because he has never gotten any further than a calm addressing of the precise issue at hand. Same for our other two as well. It sounds very strict, but it's rarely used, and our kids hear us addressing them respectfully as the norm and praising them all the time. They absolutely trust what we say when it's important, and we give them tons of freedom to be silly and crazy at all other times.
If you teach them a consequence comes after one warning, you will only need the warning. Yelling is NOT a consequence, it's an adult tantrum resulting in not handling the issues at hand and letting them progress.
Most of all, in addition to rare effective discipline, kids need to act well because they come from a happy, loving home with respect for their parents. It's very hard for them to grow into that feeling in a home where people don't follow through on boundaries or expect good behavior and then when everything is way out of control they start screaming and ruling by visible anger.
You should call a house meeting, explain the new policy, let everyone know what your expectations are and that you will follow through right away, not wait until you're mad enough to yell. Offer treat for X amount of time for the right behavior (we never need to do that, but you can)Then follow through until they believe you. Then they can control their own behavior just like you are controlling yours. Either way, you're going to have to discipline. Just act BEFORE you get mad and cut the anger and yelling.
You're an adult. You can yell SOMETIMES, but if you lower the chaos at home, you won't need to, and doing it rarely is a lot more effective than doing it all the time.