You don't need to buy new cables, unless the ends are too frayed, or they're too short after you cut off the ferule.

The cable and housing that was running from the left brifter now runs from the right brifter and vice versa.

You will have to take the cable loose at the caliper, and completely unthread it from the brifter, and you'll have to take the housing off from under the bar.

The long housing and cable that used to run from the right brifter to the rear brake, runs from the left brifter to the rear brake.

The short cable and housing that used to run from the left brifter to the front brake now runs from the right brifter to the rear brake.

No need for new cables if they're in good shape.

And you don't switch which cable is attached to which brake, but you can't get the cable out of the brifter, intact, without taking it loose fromt he caliper.