The company might or might not need to pay taxes differently on its end.
On your end, you (the individual) are required to file an income tax return with the state you reside in. If you lie and say you are unemployed that's very obviously fraud.
Whether the company you work for is willing to deal with any extra tax paperwork that might be required if you relocate is of course an open question.
No, there are residency laws in each state. I'm not sure what happens when they conflict. For example, camping for 1 month each year in WA isn't going to get you out of paying CA income taxes if you own a home there that you live in the other 11 months of the year.
On your end, you (the individual) are required to file an income tax return with the state you reside in. If you lie and say you are unemployed that's very obviously fraud.
Whether the company you work for is willing to deal with any extra tax paperwork that might be required if you relocate is of course an open question.