If my family physically moved into another state and stayed hotel and Airbnb to buy a house, could we say we are residents of the state from moving date?

Each state has their own specific residency requirements. They also have their own specific requirements to not be a resident, or to terminate residency status. So it depends on the states involved here.

Basically, you may end up filing a part-year residence tax return for each state, depending.

Yes, it makes no difference if you were renting or moved directly into your new home.  Your date of residence generally, is the date you first moved into the state with the purpose of making it your new domicile.