2008 light switch - how to replace light switch

What type of headlights do you have? HID?
Are you saying that the front and rear lights go off and on at exactly the same time? In unison?
The passenger side lights work normally all the time?
How old is the 12V battery? Original equipment?

 

I suggest that you need to test the switch before actually replacing it; might save a good bit of money.

There likely is a bad connection that goes open when it heats up; might be in the switch, might be somewhere else.

 

From the circuit diagram, one can safely conclude that the problem is not with the light switch (i.e. on the end of the stalk) as there is no separation of left and right side wiring at that point. The rear lights have no individual left-right wiring at all, i.e. they are in parallel. The head lights and tail lights are on separate fuses. There are separate left and right fuses for the head lights, but those have no connection to the rears.

Note that the license plate light is powered by the RH (passenger side) tail light, so check that for another clue.

We have seen plenty of intermittent head lights over the years, and a smattering of intermittent tail lights. The former are generally due to the need for bulb replacement. They can sometimes be encouraged to stay on for a while by switching them on and off until they start. The tail light problems can be due to the need for a replacement LED, or due to a corroded ground connection which can be found under the liftback latch, interior side.

Another area that could tie these two together is poor connections to your new battery, which can cause voltage fluctuations that might interfere with HIDs coming on. Check the bolt to the chassis for tightness. Make sure the battery is well charged.

There is nothing for it but to get out a DVM and start checking voltages. Start with the tail lights and see if there is voltage to the plug. I think there are two problems, and the fact that they are happening together is mostly fortuitous.

 

Nice theory. But I wouldn't call two different problems ALWAYS manifesting themselves at the same time "fortuitous".
I think that is unlikely unless the two problems have a common factor.