If your ex and the children are living in the house you all lived in, I believe that you cannot enforce a sale without her consent as the house is in joint names and you cannot force them out onto the street. You say that you have two mortgages. Did you take the second out after separating or was this in place before the split? I would be surpised that a lender would give you a 2nd mortgage if you couldnt afford it.
Can your ex afford to buy you out? That could be an option. If you are paying the mortgage on the family home you shouldnt have to pay maintenance on top.
As for new partners I have heard that the courts can take the new partners income into account.
You should probably get some legal advice as I am not an expert and I am only relating to my personal experiences. I hope you get it sorted amicably - its so much easier if you can - on all of you. Good luck.