I am considering a job change. I would live in Texas but work remotely for a Washington based company. How would this affect my taxes?

Neither Texas nor Washington has a state income tax.   You would just continue to file a federal tax return like you have been doing.

Nether TX nor WA has a state income tax.  But did you mean Washington, DC?

But note that, if you are a W-2 employee, you cannot deduct any home office or related expenses.

Yes Washington state. They also have a separate office in Idaho which could be where I would be located if I were to be onsite? Do you know if this would also make a difference? 

You would owe ID tax on your wages for the days you worked in ID.  You would file an ID non resident return. 

See http://money.cnn.com/2013/06/28/pf/taxes/business-traveler-tax-threat/  

And if I was given option to work from home in Texas throughout the year how would this affect taxes? 

If you  work from home in Texas throughout the year, you would pay no state taxes.

What about if it WAS DC. What would be the tax implications if I choose to establish domicile in TX but work in DC?

You would have to pay tax to Washington DC on the income you earned while working there.