Hi!
My husband and I moved to Oregon in June. Before we moved we lived and worked in Idaho and paid state taxes there.
Upon filing our taxes this year, we discover that we are being charged hundreds of dollars by the State of Oregon for income we did not earn here nor even lived here when we earned it.
We went to a professional tax person and he said there was nothing we could do about it... (this guy didn't seem to be the worlds best tax preparer though considering it took 7 hours for him to fill in 2 W2's because he kept forgetting to enter one before he pressed "enter")
When I look at the Department of Revenue site for Oregon it clearly states that Part Year Residents only need to pay taxes on income earned while living in Oregon or income that came from Oregon sources.
What do we do?
Is this possible that we will have to pay state income tax twice?

Yes - you pay OR income taxes only on income you earned while an OR resident. I think your confusion may be that the OR income tax form (Form 40P) requires you to report your full year's income, so it may seem like you're being double taxed. The way it works is:

(a) you report your full income for the year (in column F of form 40P)
(b) you show how much of that income is actually OR source (in column S)
(c) you calculate the percentage of your income which is OR-based (line 39)
(d) calculate a preliminary OR tax as if it was all OR source (line 49)
(e) Finally, multiply the preliminary tax calculated in (d) by the fraction calculated in (c) to determine the actual tax owed to OR (on line 50).

Bottom line - you pay OR tax only on OR income, but at a tax rate that is calculated based on your full year's income.

BTW, the part-year resident tax return for Idaho uses the same technique - note how the Idaho form (Form 43) requires you to report your full federal income on line 37.