There is no such thing as a part year federal return based on moving between states. TurboTax is not assuming your income is different than what you reported on your W-2.
What has happened is since you did not enter all of your income for the year, your total tax for the year had not computed yet when you observed a $350 refund.
If you would like your employer to withhold more tax, so that you do not owe any money next year, you can request increased withholding. The IRS offers a withholding calculator, here: Tax Withholding Estimator.
You can use the tool to help estimate how to revise your W-4 (what your payroll department uses to withhold tax from your paycheck).