After conversing with a friend who works in the State House, and doing some research of my own, it appears that the expanded domestic partnership rights bill will be approved by the House and sent to Governor Christine Gregoire for signing into law.
The Senate passed the bill (SB 5688) on March 10, 2009, and issued this report. The House issued this one. The bill allows for virtually the same legal rights for same-sex couples as those provided to opposite-sex ones, short of using the word "marriage". Furthermore, another high point of this bill's life in the legislature is that it was amended to not include it on the next general election's ballot. That is to say, it will not be put to a vote of the people (such as we saw happen, albeit in a different context, in California with Proposition 8).
The House Judiciary Committee will be including the bill in their hearings tomorrow, March 23rd, as well as in an executive session on Thursday, March 26th.