I currently work and live in Texas. I am moving to Arizona. However, I will continue to work for my employer in Texas from Arizona. How does that effect my taxes and withholding? My paycheck is paid semi-monthly through PAYCHEX. I am a salaried employee.
THANKS!

Your physical location dictates which state income taxes laws apply.

Hence, the day you move to Arizona is the day your earned income becomes subject to Arizona state income taxes.

Make sure your employer know this so he/she begins to withhold and pay the Arizona state income tax. Give him some warning, because he may need to open an account with the Arizona tax department, and that can take a few weeks.

I have a business in Washinton State. I plan to hire an employee in Texas. What payroll witholding/reporting obiligations will I have in Texas?

Like Washington, Texas has NO state income tax.

So the ONLY tax you need to be concerned with is the unemployment tax on the state level, and you can just pay on the federal level and ignore the state requirement.