In 2004, I lived and worked for 8 months in NJ, then moved to NY, but still worked in NJ for the last 4 months. I had a "desk audit" performed on me, telling me I owed $7K in unpaid NY income taxes. Since I worked in another state for the entire year, and paid NJ income tax, can I get out of this?

You should have filed a part-year resident income tax form in NY for the 4 months you lived in NY in 2004, as well as a part-year resident form for NJ. On the NY form (IT-203) you should have reported ALL your income for 2004, but you actually get taxed by NY only on the income you earned while a NY resident, and the amount of tax you owe onthat income is reduced by a credit that NY gives you for the tax you had to pay NJ on that income. Bottom line is that the actual amount of tax you owe is probably pretty small (basically the difference between NY's tax rate and NJ tax rate on the income for those 4 months). But if you failed to report ALL your income to NY, or don't show the credit for taxes paid to NJ properly, they will assume that you owe more taxes than you really do.

File NY tax return. Report income that you earned while in NY. Claim credit for taxes paid to NJ on that income.