What date do I use for residency? I recently moved, lived with family for a couple months then bought a home. I got my driver's license before. Date of closing? License?

Did you live with family in another state?  The dates you were physically living in the state whether or not you owned your home is the dates for residency.  The day you got your license and voters registration are not relevant.  If you lived with your family for a couple of months in the same state those months count too.  

Thanks!