It's a great question. Washington state tenant laws supersede sales, so your lease will survive the closing of any real estate purchase and sale (short sale or not). The new owner will have to honor the terms of your lease. Either the seller or the new buyer may try to "buy you out" of your lease by offering you an incentive to move early, but you are not required to do this.
To be sure, you should consult an attorney.
If you really like the house, though, why not offer to buy it from the current owner? You don't provide enough details on the owner's financial situation--and if it's very difficult, she may not be in a position to provide owner financing. However, if she's just thinking: "My husband died and I don't want the home," you could offer to buy it from her.
Now, if it's a short sale, then presumably she owes more on it than the property is worth. And if it's a lot, then it wouldn't make sense for you to overpay for the property. But if it's close--or, better yet, if there's some equity in the property (in which case she really wouldn't have to do a short sale at all), you could buy it from her with owner financing (maybe on a land contract). Or acquire it in a "subject to" transaction where you'd receive the deed and start making the mortgage payments. There are a number of variations of that strategy, but you get the idea.
Be sure to talk to an attorney if you do decide to go ahead with one of those options.
Hope that helps.