The reason you're shouting is because you've lost control, so you need to regain control of the situation. (Sounds sooo simple :o)
Note that you can't control everything they DO. You can only control other things - for example reducing their options, your own reaction to their behaviour, your communication to them, etc. By the same token: You can control things like what options they have to eat, which shoes are available, what time you aim to leave the house, how much there is to get ready, the route you take to school. You can't control - their moods, their actions including whether they put the shoes on, traffic, weather, availability of parking. It's useful to separate things out like this, stops you from getting angry at the things you can't control.
If you are not a morning person the biggest thing you can do to help - though the thought is torturous - get up earlier, preferably about 20 minutes before the kids. Why? Because it gives you a window of peace to come to before you have to get in role. Make yourself a coffee and sit scrolling through FB/open a window/jump in shower/whatever will wake you up in peace.
Reduce the amount you need to do by prepping as much as poss the night before so you're not stressing - lunches, bookbags can be done. Uniform laid out to aid with dressing and avoid any last minute panic searching for socks, shirts etc.
If 6yo can't be trusted to dress unsupervised don't expect him to - worry about why that might be at another time. For the moment, in the mornings, just deal with what you have. Meet the child where they are rather than trying to go with where you think he should be. Get it functioning and then if you want to you can work on small incremental changes (not leaving him on a totally different floor for 10 minutes, that's too big a jump). So get DC in one room to dress - you help the 3yo while the 6yo does his own but in the room with you. Try to get them to see if they can beat a timer - don't get them to race each other as it causes arguments.
You might be trying to squash too many tasks into too short a time. If it takes longer than 10 minutes to get dressed fine. It might help to do a "mock up" morning (or other time sensitive tasks) at a non stressed time and see how long each individual task takes so you have an idea of how long you need to be leaving for each bit (with 5 min contingency between each task). Again meet child where they are - a child isn't going to complete a task (even a simple one) in as short a time as an adult.
I used to find TV useful for mornings in that - if DS1 ready by certain time, TV could go on and one episode of Peppa (on Milkshake) could be watched. When Peppa was finished TV had to go off as it was time to go. If he didn't finish getting ready in time we had the 10-15 mins (can't remember now) that Peppa usually lasted as back up/extra time. That was genius because it was essentially something they couldn't argue with because I couldn't change the TV schedule.
Divide (tasks) and conquer - if you have a partner who doesn't leave for work 2 hours before you need to leave for school, get him to take over supervising or preparing some of the tasks.
Make your boundary separate from the actual line, add a "safety zone". For example. If you know that you need to leave by 8.40 and you will absolutely be late if you leave after 8.43, don't wait until it looks like you'll be leaving at 8.41 to get really cross. That's not enough grey area. Instead, make your boundary be 8.25. If their behaviour is causing you to need to leave later than 8.25, that's the boundary for behaviour management of your choice to kick in. Does not need to be shouting/telling off, could be loss of star chart or some privilege en route, could be a different/positive technique, could just be you chivvying them along to remain on task for an 8.25 exit. The difference is - you'll be able to stay calm, because you know really if you leave at 8.28 or even 8.30 you'll still have loads of time to get there. If you have your boundary the same as the actual time you NEED to leave then you've got no safety zone when they push it and you'll tend to go into panic mode and think "But I NEED to get them moving" and then you'll shout etc, which doesn't work anyway, and just gets everyone really stressed out.
I've focused on mornings but apply same principles to any situation:
- Remember what you can control vs what you can't
- Meet the child where they are until the situation is manageable, then make small incremental changes
- Build yourself in "breathing space", as in safety zones between parenting boundary vs actual, OMG panic boundary, also in allowing extra time, finally in general (20 minutes to sit down with a cup of tea perodically is important)
- Divide and delegate (to husband, to "past/future you")