Send the employer a copy of
NJ-WT? Although it has occasional cites to NJ statutes, I don't see specific cites for non-resident withholding and reporting, just the instructions asserting the requirements. Thus I don't know if there are any exceptions for incidental employment in the state (e.g. paying an employee to give a talk at a conference, when the employer has no other business connection to NJ).
I'd expect something similar for LA, but I'm less familiar with their tax system and web site.