I'm moving from California to South Dakota. Is it legal to transport my shotguns in my car?

There are no overly restricting states between CA and ND so there shouldn't be any problems. The Safe Passage clause of the Firearm Owners Protection Act applies if several conditions are met when transporting firearms. 1) The firearm is not accessible to the driver or occupants (locked case in the trunk). 2) and the ammunition is unloaded and kept separate from the firearm.

Good rule of thumb regardless of where you're going is to put the guns in the trunk cased with a lock and place the ammo in a different compartment!

I have travcelled all over the western Us and have never had any problem transporting firearms. I drive a pick-up so carry my long guns in the back seat, under a cover. Unloaded and no ammo in reach.
Enjoy SD,, this is gun-friendly country.