Per FICO themselves, there is no difference to your credit score between foreclosure, short sale or deed in lieu. See link.
Best I can tell the only difference between these options is the length of time one will have to wait before qualifying for a Fannie/Freddie-backed mortgage again.
This is a critical step in the process. Secondly, they are geared up to the process of working a short-sale and communicating with the lenders on a continuing, consistent basis. So you really need to pick a Realtor/Agent who has successfully closed a Short-Sale transaction.
But in your case, with an approval already given, you need to sally forth and get it done. However, I would look at the approved lender "pay-off" and make sure it is now realistic to the marketplace. These take so long and you may find your home/pay-off OVER market value at this time! So revise your market analysis, talk to your Realtor/Agent and get the deal done. It will be better than a complete walk-away foreclosure.
Your agent can advise you best in regards to your specific situation. I agree with others that a short sale will be better than a foreclosure from a credit standpoint. The short sale process is obnoxious. Absolutely obnoxious. The game has no real rules and certainly does not run by logic. In my humble opinion the loss mitigators have large egos and are incredibly short sighted. They seem to take no responsibility as a representative of the company who made the loan in the first place.
With that complaining aside, I agree to list that house again at the "approved" price. Have your agent call back anybody that was interested before but didn't make an offer. The goal is to get a second offer asap. In the mean time, try not to worry about it. There is not much you can do and your agent should take care of the details in order to spare you as much stress as possible. I recognize it may hard to let go, but save yourself the stress and try to think about how good it will feel to finally be rid of the burden.
Many buyers are shying away from short sales and many agents are not showing short sales because of the expected length of time for the lender to approve the sale. This is especially true in Sonoma County where we are still in a declining market. Now that you have your lender approval, I would definitely put your home back on the market and have your agent put in the listing that the bank has approved the sale for your list price. This is much more appealing to a buyer as it pretty much guarantees that the sale will go through. Having completed many short sales this year, I am seeing an increase in the lenders cooperation to get these deals done and avoid foreclosure. Have your agent check the comps to see if your current list price is still appropriate for your area. And once your agent does put your home on the market, they should launch new advertising indicating your recent lender approval.
Good luck - I expect that with your lender approval, and if you are priced right and in good showing condition, you will see another offer soon.
Who is your Realtor? I would think that they would be in the best position to help you.
The short story here is that a short sale will ding your credit score 80-120 points on average, a foreclosure 150-250 points, meaning you won't be buying anything for a long time.
If you would like I would be happy to refer you to an expert in short sales in Santa Rosa. You can contact me via my Trulia profile and I'll do my best to help you. They can analyze your situation and then you can decide what's best for you.
From the lender's perspective they would rather have a short sale than foreclosure. It is simply a matter of money. So if I were you I would still try to sell.