Hi Paul,
All understood now. You could sell your car, use the money to declare yourself BR and to buy a cheaper car thereby losing the debt liability that you have. The joint loan with your partner would then be her responsibility but you could contribute. Rather than raising a mortgage, which will be difficult anyway, stay where you are and have a reasonably debt free life. I presume that you are looking to buy out her ex's interest in the property.
If you only pay off one of the cards, how long is it going to be before the others place a burden on you.
Hope this helps. Send me an email if you want.
Best wishes,
Steve