Get the house valued by three different estate agents. Deduct how much you owe on the mortgage and what is left (the equity) is split 50/50 unless you had a prior agreement when first bought.
Get the house thoroughly cleaned, ie move furniture and wash skirting boards, clean inside cupboards etc. Whilst doing this you can declutter at the same time.
Do any small jobs that you've put off such as skirting boards falling off, misty double glazing, broken cabinet doors, dripping taps. Don't repaint as it's a waste of money generally unless the walls are horrendously scuffed.
Mow the lawn, cut shrubs back, sweep paths/drive and throw away broken pots.
Get on Rightmove and see what you can afford and where. Keep tweaking mileage and prices and downsize your expectations.
Get onto CMS and start the application. Go on benefits checker to see if you can get any help with childcare or housing costs.
Be prepared to do it all with zero help. Better to be pleasantly surprised rather than have even more resentment and anger.
Finally, good luck. You've got this 
EDIT - the pp above seem to think you are married. I'm assuming you are not as you said DP. Can you clarify?