If your legal residence was GA for the entire tax year, you are not subject to Tennessee's Hall Income Tax, which is a tax on interest and dividends.
With RSUs, you are taxed when the shares are delivered, which is almost always at vesting. Your taxable income is the market value of the shares at vesting. It is considered compensation income subject to federal and employment tax (Social Security and Medicare) and any state and local tax.
Therefore your RSU's will be taxable at vesting at both the federal level and by your home state of GA. You won't owe any tax to TN.