Change your address through payroll to Texas, from then on you're paying TX state tax (aka nothing). You will be accountable for RI tax on anything prior to that. Technically speaking, you must be working in Texas at that point, so make sure you have a residence in TX (i.e., purchased home or lease) at that point and start moving all your stuff over. When you file taxes the next year you'll have to file in RI for anything up to the point you moved. Sounds like you have an accountant so he can figure that out for you, it's not even hard doing it yourself.
I've lived in 5 different states, there's not much to it. If RI dings you for car registration, property tax, whatever, you'll be responsible for that as long as you have the asset located there... getting Texas plates requires inspection, so that's a defacto point in time to tell RI to fuck off for auto taxes.