Yes, it is correct that your compensation from the exercise of non-qualified stock options is included in box 1 of your W-2 as wages.
Yes , you need to report your 1099-B from E-Trade on your tax return. You will need to adjust the cost basis when you enter your 1099. Your actual cost basis per share is the value of the stock at the time you exercised the options. More than likely the stock basis on the 1099 is the option strike price that you paid. But the difference between what you paid and the value of the stock (the bargain element) was added to your W-2 as income, so your cost basis is the value of the stock when you exercised.
California considers a portion of your income from the options as having been earned when you were living/working in California. The options grants were probably awarded while you were in California, so if you leave, they follow you and tax you.
As an Oregon resident, 100% of your income is taxed by Oregon, even the income taxed by California. However, to prevent double taxation Oregon will give you a credit for the tax you pay to California. As you suggested, Oregon will only credit you for the amount they tax that income, not the amount you paid to California.
You don't have to do anything special in TurboTax to properly report this. When you enter the 1099-B from E-Trade, do not import it, enter it yourself. You will be asked if your sales involve company stock, answer yes and the program will guide you through entering your trades and adjusting your cost basis.
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