I think you'd benefit from independent financial advice. An hour of honest advice about your choices and their potential impact would be worth it.
My gut reaction is that you shouldn't sell, as it's your escape route and it's a short term solution to an ongoing problem. What will happen when you've paid off your debt, you no longer have an additional source of income coming in, and no potential of selling a property again? However, that may not be financially best. It would be good to look at short and long term ramifications of:
- Sell the flat and pay off the debts
- Don't sell the flat, but find a way to stick to a budget and pay down the debt over a defined period of time
- Sell the flat but invest the capital and pay the debt down over a period of time (but in larger payments than you would be able to do without selling the property).
You both really need to talk openly about money without sniping at each other. 'Look, this is where we are; let's decide together on options that work. I'm uncomfortable selling the flat, as it provides a long-term safety net for us, but what if we looked at options x,y,z?'
You also BOTH need to pay for the children's costs.
And think about long term rather than immediate needs and benefits. You can't continue working 2 days a week. You could look at getting an evening job (I appreciate all jobs are very hard to get nowadays, so I understand that may not be possible) and, depending on your skill set, it could be something done at home. Or you could plot out your finances for the next 3 years and say, for the next 12 months, things will be tight, but then both kids will be in school and I can work more hours etc, or look into the cost of upskilling, or consider DH's promotional opportunities or chances of getting a job on a higher salary elsewhere.
You know this last bit already - you both need to be better at planning and stop being frivolous with money. You forgot water? Go without, find a bubbler or do as someone else suggested, and always keep an empty bottle in your bag. Don't forget wipes. Have a 'go bag' always packed, so you know if you're in a hurry, when you pick it up it will have spare nappies (if necessary), wipes, tissues, an empty water bottle, a few muesli bars or packets of dry food that won't go off, and a book and colouring for the kids. There's no excuse for being unprepared when you know you can't afford it. Don't pick up books and just buy them - take a photo and find a cheaper option online or borrow it from the library.