A short sale almost certainly will negatively impact credit scores because you are settling the mortgage debt for less than you agreed to pay originally. I say almost certainly because in rare cases a lender may report the debt as paid in full and forgive the remaining portion of the debt.
While it is possible that the lender would forgive any remaining balance, the words "short sale" will not appear on your credit report. Rather, the account status accurately should be reported as "settled," which means paid but not in the full amount that you originally agreed to repay, just as your lender described.
Because it is a mortgage, the impact of settling the account will probably be significant. Just how significant, though, depends on your unique credit history.
A short sale would have less impact on the credit scores for a person with no other negative information than for a person who had late payments or collection accounts in their history. In either case, though, the damage to your creditworthiness, and therefore your credit scores, will be substantial.
If your objective is to get rid of the debt so you can quickly apply for new credit, a short sale probably won't help. That should never be your goal, anyway.
Rather, you need to think about your long-term credit health. A short sale is one option that may help you get out from under debt you cannot manage. Once you are free of that unmanageable debt, you can begin to take steps to rehabilitate your credit history so that in months, or maybe even years, down the road you can obtain the credit you want and can manage in ways that work to your advantage.