It's OK. It's a horrible feeling but it is only money. At worst you could go bankrupt and start again. But it probably won't come to that.
Two cars on top of a mortgage that eats up 33% of your income, plus having kids - those are huge expenses. Look hard at whether there is any way you can go down to one car. Can you transfer or change jobs to be nearer to home? Use public transport or a bike? Lift share with a colleague? Walk to work? Pick each other up?
For the kids' birthdays and Christmas, swallow pride and get onto Freecycle. If your own area isn't very lively join one in a wealthy area nearby that is. We have given away/received good as new or brand new bikes, toys, books, DVDs, musical instruments, sports equipment. You can top up with a few small things from Poundland. Or come on here and see who's offering giveaways. People on MN love to give away good stuff to people in your position.
Tell adults you're not buying this year, but agree with DH to get each other something small and fun so you have some fun and surprises on the day too.
Make a meal plan for the week and only buy food on it. Avoid takeaways. Take coffee in flasks and sandwiches from home to work.
Make sure savings you've made are used to pay off your biggest debts. Start with those that demand highest interest rates.