Q. Is this my responsibility to file a nonresident tax return for every state we visit, and I work from?
A. Simple answer: Yes.
You'll need to determine the rules for each state. There may be some exceptions for short stays, but probably not. For quick links to State Filing Requirements: https://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-return/need-file-state-return/L446Rv...
Q. I am afraid my employer is not going to allow me to work and live this lifestyle anymore because of the tax implications?
A, There are no implications for the employer. They have no obligation to withhold state income tax for the states you intermittently work in. The responsibility falls on you.
Q. Is an Itinerant worker responsible for state income tax?
A, Yes, but few do.
See http://money.cnn.com/2013/06/28/pf/taxes/business-traveler-tax-threat/ Since you live in a state without an income tax (Florida), then you should file in the work states. You can't use the "it all comes out even" rationale for not filing.
