Q. Are they correct in stating for him to check exempt on the Oregon W4?
A. Yes.
Q. Will this keep us from having to file a nonresident for Oregon?
A. Yes.
Q. How do I handle NM withholding?
A. There is no way to have NM tax withheld from his pay. You'll have to make quarterly estimated payments to NM. See https://www.tax.newmexico.gov/individuals/file-your-taxes-overview/estimated-payments/
You can prepare 2023 estimated tax payment vouchers in your 2022 TurboTax (TT) state software. I'm not specifically familiar with the NM program, but it's usually at the screen titled "A few things before we wrap up your state taxes", under "Other forms". It may be a screen "Do you want to pay NM estimated taxes for 2023".
