I had loads of trouble after DD2 was born, but DD1 was only 2 at the time. My children are now 4 and 2, but I also had PND. Are you on your own, or can you get help from your partner or another adult? I know that the root of all my problems are to do with attention.
Can you give DS2 attention while DS1 is at school? By that I mean spend 30 minutes a day just playing with him/reading to him etc and not trying to do your jobs?
Can you stagger bedtimes, so that DS2 goes in bed first, and then spend 30 minutes with DS1 while DS2 is in bed? It sounds as if they are in the same room, so I appreciate this might be difficult, but if DS1 was downstairs, DS2 wouldn't have anyone to play with, so might be settled by the time DS1 went in.
Do they respond to reward charts? If I come up less than three times tonight you can choose a treat out of the bag when you have had your breakfast and got dressed in the morning (that way they also have to get going in the morning if they are going to have their treat before school - 2 for the price of 1) if they dawdle they still get the treat, but when they get home.
My treat bag mostly has mini boxes of smarties, play dough, lollies and occassionally small toys, rubbers etc (and hairgrips, but that's no good to you)
Try and notice when they are being good. Praise, praise praise when they are. My friendly child pshchologist says 8 good things for every one bad.
Try and pick a day when you think you have a good chance, maybe when you have a grown up helper about and say to your kids (you might need to do this one at once)
'I'm sorry I've been shouting so much recently, and it isn't very nice. From tomorrow I'm going to try and stop shouting so much, and you can help me by doing XYZ' try and say 'look after your room' rather than 'not wreck your room' so you are making it positive.
If we can manage to have a nice day tomorrow, then on Sunday we will....
(go swimming, go to the park whatever)
My DD1 was only 3 when we did this, but she really got that I didn't want to shout, and she didn't want to be shouted out. It's been a long slog, but suddenly the naughty step has started working for us. Even if it doesn't make much difference to their behaviour, it was a lot nicer being me when I stopped shouting constantly. Actually her behaviour has improved vastly, but is still a long way off perfect (she used to be very, very violent)
Try and sleep now, and see what everyone else has said by tomorrow.