So my question is, can my full income be taxed by two different states if they both have a state income tax?
If you never worked or lived in New Mexico then you should have filed a NM non-resident state income tax return showing no income earned in the state and all state taxes withheld to be refunded to you.
Contact whomever in NM sent you the notice and explain your work situation to them. This should be a an easy issue to resolve.
There are a few states that tax telecommuters in other states. NM is NOT one of them. But the fact that your employer withheld NM tax and indicated NM in box 15 of your W-2 tends to make the the NM tax people suspicious.
You should have mailed in (not e-file) your non resident NM return and attached a statement of explanation. An additional letter/statement from your employer would be helpful.
At this point, I agree with @DoninGA , you need to write to the NM Taxation and Revenue Department with an explanation. Try to get your employer to stop withholding NM tax.
Another issue is that, if your employer is not withholding SC state taxes for you, you must make quarterly estimated tax payments to SC. See this:
Your income is 100% taxable by your resident state of SC.
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