OP, you clearly need some help so I hope that you can source some. The school were great at helping me when DC went through a really bad patch behaviour wise.
I do agree with PP, and please don't take this as criticism, but you do need to get your DC to sit up at the table and eat. If DS was allowed to roam around like DD is now, then of course he is not going to sit still in Burger King or wherever.
Actually, it doesn't matter if they sit on the floor and have a picnic, but the main thing is that they sit whilst eating. It is a very basic lesson to learn in life. Your DD will soon sit down if she realises that there is no food unless she does.
I had a lot of help from professional agencies and it was recommended that me and DC sat and ate together, as it is time together.
Magic Time was also recommended, so just 20 minutes together reading a book or playing a game, but every single day. You do need help to do that though, so that someone else looks after other DC while you spend time with each one separately.
DD will grow up to follow DS's example and so on and so on so you really need to get some help now.
The Family Support Worker who helped me, stressed that they need firm boundaries, firm rules set in place, only say things that you can follow through, ie , walk beside me or we go home and if they don't then you do go home.
Also acknowledge that they are angry/upset or whatever and give them a hug and say I understand you are not feeling happy, what can we do about that? It is not a matter of giving in, but acknowledging their feelings, then calmly explaining why they can't have/do whatever. It can be very hard to do when you are at the end of your tether though! but it did work with DC.
School will help your DS, so don't delay him whatever you feel about him being too young. They will soon pick up on any problems too.