The difficulty with finding death records is that, unfortunately, people can die anywhere. There's no guarantee you'll find them in their hometown. They could die at a hospital across a city or county line from their residence; they could die while visiting a relative, or they could die in transit. So one possibility would be to imagine where Sarah might have gone, if she had visited or gone to live with other relatives. Do you have an obituary for Joseph? Have you identified all the survivors listed there?
We can guess that Sarah died sometime after census day in 1910. The Tacoma City Directory which has a 1911 date on it was probably 'put to bed' sometime in 1910 -- we don't have any evidence about when the 1911 Directory would have been published or distributed. Sometimes if you read the front matter in the directory, you can find internal evidence that will provide a clue about what month the directory was distributed, but usually I just try to remember that the information in a directory is likely to be from the year before the date on the title page.
The FamilySearch Wiki article Washington Vital Records says that statewide registration of deaths began in 1907, but general compliance didn't happen until 1917. So -- assuming for the moment that she did die in Washington state -- this means to find a death record, you might have to go to the county where the death took place -- not a trivial task when you don't know where that might be.
Sources of Genealogical Information is a checklist of other records you might check to find a date and place of death.
Joe Beine's Online Washington Death Records & Indexes is a guide to resources available to search for people who died in Washington State.
New York State Records
There are a couple of reasons that researching vital records in New York State is not as straightforward as it might seem. First, not all the New York State records can be found in one place. New York City keeps its own records, so anything post-consolidation needs to be looked for in NYC. Albany, Buffalo, and Yonkers also keep some of their own records. A good explanation of all the exceptions can be found in this guide to genealogy records and resources from the New York State Department of Health.
Another problem is, if she died in New York State, and you are researching from outside the state, you may have difficulty finding her via an online search.
The FamilySearch Wiki Article, New York Vital Records has a list of online resources. Their article says:
Most online resources for New York Vital Records are indexes. The
official New York state vital records index is held on microfiche at
select libraries in New York. According to state law this index cannot
be copied. Therefore it cannot be put online and cannot be viewed out
of state. After locating a person in an index always consult the
original record to confirm the information in the index.
Because of this, you may have less of a chance of finding an entry in an online index for a death in New York State, compared with some of the surrounding states like Massachusetts, whose governments or archives provide a searchable index.
The New York State Archives has research guides for genealogy researchers. See Birth, Marriage, and Death Records for a list of which libraries hold copies of the microfiche index.
Joe Beine's Online New York Death Records, Indexes & Obituaries provides a guide to many searchable indexes, starting with statewide resources, then organized by county.
If you're feeling adventurous, for New York State newspaper research, try FultonHistory.com.
