Getting a house built – smartest place for transitional housing/apt while we wait?

I recently sold my house and am getting a home built in Richmond, TX. I need a comfortable place to stay for a few months. What do people normally do? AirBnB a place for a month or two or find an apartment to rent month-by-month? I think with AirBnB i don't have to worry about utilities but are there any apartments that do month leases without having to worry about wifi and utilities? It's been awhile since I've rented and just need some insight what people normally do if they need housing for a month or two (or three?!) while their home gets built. Any advice is appreciated.

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BTW My budget is $1200-3000 per month but would def like to save as much money as possible while still being comfortable (me + wife + 1 child)