If I’ve been working as a shift manager at a restaurant for 2 1/2 months and they just now started paying me the shift manage wage, am I owed back pay?

In Washington State, many of the terms of employment can be oral in addition to written terms.  I suggest you look at any written employment documents first such as any employment handbook or contract. 

Washington is not particularly protective of employee's rights so you may want to ask yourself if this is really worth causing an issue over with your employer.  It wouldn't feel like much of a win if you got a small amount of back pay, but lost your job for "other" reasons.  It might not be legal to do, but it sometimes happens....

As always, my comments are only applicable to Washington State and are not a substitute for getting competent, local, and more comprehensive, legal help.

You would have a claim for the back wage differential if:

1) You had an employment agreement or contract which guaranteed you the extra money; or

2) To take the job as shift manager, you gave up some other opportunity or did something else to your detriment (e.g. gave up a second or outside job), and your employer knew or should have known you'd do that at the time it promised you more pay, and the reason you did this was the promise of more pay.

In either of these two situations, you would seem to have a legal claim for the extra pay.

However, barring the above, you probably do not have any claim. The law does not require employers to provide pay raises with promotions, and allows employers to freely decide what to pay their employees. Only when there is a contact or if the conditions of "promissory estoppel" (option 2), above) are met is an employer obligated to pay more money.