Your questions may need more advise than can be provided on a public forum by Realtors. You may very well need an attorney.
If you are responsible for the property (on the loan, on title, living there), it is in your best interest to protect your credit and resolve the outstanding issues. Options may include selling the property, may include keeping it and taking in a renter to offset the expense, or may include signing over the property to the other individual. Please do not act on any of these options without proper counsel from someone who is knowledgeable about your circumstances What I proposed as possbilities was provided for informational purposes only. I do not have enough info to make a recommendation. I do hope the info herein may help you in furthering discussions that will lead you to a good decision for you.
Have you thought of getting a roommate, someone to help share the living expenses. If you are a mother, I'm sure you have single mother friends. Try talking to your friends about it. Also, you could rent out a room or possibly rent the entire home. Just do whatever you can to avoid foreclosure... legally!
I wish you the best of luck!
After a breakup of this magnitude, emotions run strong. Add to typical emotions the financial strain of a home and mortgage and the it can be easy to cross the line into all out war.
You have to maintain perspective on this situation. I'm not sure if both of you siged as responsible parties to repay the note, but the lender holding this note is not interested in your emotional well being. They are vitally interested in how you plan on settling the account you have with them for the house.
I speak to the issue of avoiding foreclosure on my YouTube Channel http://(www.youtube.com/themortgagecoach). Click on the video labeled avoiding foreclosure and listen to the suggestions I give for people in your situation.
It is regretable that you are in the situation you are, but let me assure you, there are ways to work it out. Most likely you will have to work a short sale with your lender. Please find a qualified realtor who knows how to work with short sales and allow him/her to walk you through this as painlessly as possible.