This is a big question for you and you really need to consult an attorney.
Don't try to figure this out by yourself! Real Estate Agents cannot give legal advice you really need an attorney for this!
First step is to identify whether or not the loan(s) on your home are recourse or non-recourse loan. If they were loans for your initial purchase then they are most likely NON-recourse loans. If you refinanced or borrowed more money after the purchase, then they are most likely recourse loans. Meaning that the lender can pursue the debt.
However, I would recommend sitting down with a real estate attorney because California law provides some gray areas regarding this exact issue. The consultation with a lawyer that allowed you to strike this paragraph may very well be worth it.
The following link provided this information:
2. If the borrower takes a loan for purposes other than the purchase of a property, and he later defaults on the loan, the lender must first foreclose upon the property to satisfy the debt, but can thereafter obtain a deficiency judgment for the balance of the loan. But any such deficiency can be recovered only if the lender uses a judicial foreclosure rather than a private sale. If the lender pursues a judicial foreclosure, the borrower will have the redemption right to buy the property from the purchaser at the foreclosure sale. If the lender uses a private sale, it cannot obtain a deficiency, nor can the borrower redeem the property after the sale. If the lender tries to circumvent all of this, it might find itself barred under the one-action rule from recovering any part of the debt from the borrower, or at best its secured debt will become an unsecured one.
Secondly, in CA if you had a 1st & a 2nd mortgage to PURCHASE a home, then they fall under the category of "non-recourse" loans and should not be subject to deficiency judgement. However, it is difficult to assess the situation without seeing your paperwork. Your short sale expert/negotiator/realtor should be able to negotiate the verbiage on the approval letter, given that it is a non-recourse loan. Many banks send out letters with standard verbiage instead of personalizing it to fit the individual state's laws.
Thirdly, none of us, as realtors, can give you advise as to whether or not you should accept their terms. We aren't lawyers or CPAs and you really need an advisor with that specific expertise to guide you. Best of Luck!