[Steve Tinker]

This sound like you have a high resistance grounding problem.
Try removing both battery connections (make sure you have the radio code beforehand) and clean them both thoroughly - including the internal faces of the clamps.... grease the cleaned external connections to stop future corrosion.
The other end of the negative terminal (-) connects to the chassis - this must be rust and corrosion free. Wire brush (emery cloth) the mating surfaces after removal and make sure the connection is tight.
The Boxster headlights are a "plug in" type which could have a high resistence connection which robs the rest of the electical system of voltage.
If you have a multi meter, you can sheck the resistence of the various grounding (negative) terminals but this can be very time consuming and if you don't have an electrical background a little confusing.
Hope this helps....