My primary residence is in Texas. I will temporarily be renting an apartment for work in Seattle, WA. Is this a deductible expense?

Perhaps, depending on the answers to a few questions:

1. Will you be on this assignment in WA for shorter than one year?
2. Will you still maintain your home in TX during the work assignment?
3. Is your employer not reimbursing you, and also not paying you a per diem?

If yes to all 3 questions you can deduct your T&L expenses as miscellaneous deductions (to the extent that the sum of all misc deductions exceeds 2% of your adjusted gross income). You would file form 2106 to document these unreimbursed work expenses.

If you meet the requirements, you will be able to deduct your ordinary and necessary travel expenses while you are temporarily performing personal services in the United States. Travel expenses include transportation, lodging and meals.Use Form 2106 or 2106-EZ to figure your allowable expenses that you claim on line 9 of Schedule A (Form 1040).