Like the other agents said below the best idea is to consult with a real estate attorney. The law varies from state to state, and Arizona is an anti-deficiency state. So basically what you are saying it true - although like in everything else there are many exceptions - like on HELOCs (Home Equity Line of Credit), etc.z
Even if they do not come after you for the deficiency your credit will be completely ruined if you just let the property go into foreclosure - so, why risk it? Lenders are also actively trying to find loopholes and they will do anything they can do mitigate their losses.
In your case you will be better off trying to do a short-sale. If done properly it will be a settlement and the bank will agree in writing to accept less money to satisfy the debt. Just make sure that your agent is experienced and you get the correct language when doing the short sale.