1. As Snipes said, you'll file a GA nonresident return & declare that none of the income is GA income, and therefore GA can't tax any of it. You should get a full refund.
2. You will file a RI resident return & have to pay a big balance due. File the GA return early so you can use the GA refund to pay RI. You'll probably also have to pay a penalty to RI for underpayment of taxes throughout the year, since RI got nothing. You probably can't get reimbursed for this from the company, since you should have known in March that your withholding was screwed up & didn't bother to fix it until now.
3. Talk to the payroll department about correcting the withholding for 2006. If the payroll dept gets stroppy, call the RI dept of revenue and give the state all the particulars. RI will send a polite "fix this or pay a big fine" letter to your company, which will then comply.