Sympathies, sounds like you've had a tough enough time so don't be too hard on yourself. Also sounds like you basically do understand how to budget pretty well, it's just a case of practically and maybe emotionally how to stick to it?
I'd recommend the money saving expert website and forums, they are very straight talking over there but no judgement for being in debt/bad with money and some really helpful advice.
Things that help me are having various different bank accounts (I agree that the Monzo pots are not quite enough separation to stop me dipping into them for things I'd hadn't intended).
I'd do one bank account which is for bills and debt repayment only. Put enough money in here to cover your fixed outgoings every month, set your direct debits to come out of it and do not touch it for anything else, ever. Don't even check it regularly, don't have text alerts or anything saying how much is in there or you might be tempted to 'borrow' if you get a text saying there's £500 in there even if you know you need to pay a £100 bill in a week!
A second account is your medium term savings - this is things like Christmas, holidays, birthdays, car service/MOT, any white goods that you know will need repairs or replacing this year, insurances that have to be paid annually, TV license - big bills that are annual or infrequent. MSE have some good tips on what to include in this, you might want to include things like a budget for school uniform/shoes for the kids or haircuts for you for instance. Add what you (realistically) want to spend on all of this per year up and divide by 12 and then on payday do a standing order from your main account every month to another account (something like an instant access saver or just another current account). Again try and be disciplined to not dip into this for anything not on the list and stick to the budget you set yourself - if your car MOT is due in January it's no good blowing all the money on Christmas for instance!
Then ideally you want a 3rd, longer term savings account, something you really don't touch at all. This is your true rainy day fund (lost your job and can't pay the rent, boiler has blown up or similar) and/or longer term savings to enable you to move house or similar life goals. Again set up a standing order on payday to this account so money goes straight out to it without it ever being mixed up with your day to day spending. How much you can put in will depend on how much you need for the medium term stuff above^ but if you could start with maybe even just £20 a month or something it will add up. When you are finished with debt repayments you can redirect that money towards longer term savings and (presumably) your really high childcare bill will eventually reduce too so if you can get into the habit of saving now you will be well set up to do more in the future.
Finally, on the how to manage the more emotional/impulse side of spending, I do think the withdrawing money in cash and when it's gone it's gone technique can work. I'd maybe try splitting your food, child activity and fun/pocket money spending into a weekly budget and try taking it out in cash (I wouldn't do a month at a time as having that much cash may burn a hole in your pocket). I do think you need to schedule in treats of some kind for yourself on a regular basis, trying to totally deprive yourself is not going to work and just like with food (or indeed other things that give you an endorphin hit like, erm drugs, gambling or sex 
) will tend to lead to a binge-shame-depression-binge cycle.
As you like your food I would try and budget for takeaway once a week (we do a nice/blow out takeaway like thai or curry on pay day, a moderate/cheap-ish takeaway like pizza or fish and chips most Fridays and a MaccieDs or other cheapo treat the last Friday of the month 
) which should be doable on your budget, and then maybe try to get into meal planning for other meals? Having a firm plan of what you want to cook each night and having the ingredients all ready in the fridge should reduce the 'fuck it, I'm getting a pizza' feeling? Do budget for some convenience food on busy weekday days and nicer foods/puddings or other treats for the weekend too - it's far better to plan for these things but then stick to it, than to try and deprive yourself totally of all nice things and go mad when you do have some! There are various helpful threads on meal planning on the food and recipe boards or I can share what I do if helpful?