Sorry you are going through this, what a horrible couple of months you have had.
My advice to myself in this situation would be that feelings need to come second to practicalities.
I would say that getting a house sorted is your priority. Find something you can afford (& would still be able to afford if your mortgage rates doubled or your income dipped), wherever that may be, and then sort schools out once that is done.
Yes, it would be nice for the kids to stay in their schools, but 15 hrs of commuting a week is costly and unsustainable, especially during winter when the days are shorter, colder and wetter.
Re your youngest, yes, speak to a GP, and also to his dad. Is there anything you can do together to help your youngest with this transition?
Re your oldest, if your oldest is at secondary school s/he is old enough to understand the reality of the cost of living crisis and cutting your cloth according to your means. Yes, s/he'll be upset to move away from friends, but unfortunately, being upset is part of life and something they will have to deal with.
Do you like your job or do you want to change? Is there a different role in the company? If so, this may help with mortgage as you will have continuity of payslips for the paperwork. If not, have you started exploring other options?