Alright...this one has me completely lost. Hopefully someone here has an idea!!

The first thing to note is, it doesn't matter whether the car is running or not.

If the car is just sitting there running, or not running, and/or the key is in the "On" or "Acc" position, and I pull back on the lever to flash the brights, the radio will lose power, both blinker arrows will light up faintly, and the small orange light on the rear defrost button will light up faintly. Also, if the car is running, I hear/feel the engine strain a little bit right when I pull the lever, and the engine jumps about 500rpm for a second, then returns to normal. I can also get it to do all of this if I put the heat/vent on high.

Right when I let go of the lever, everything comes back on and acts normal.

This has happened sitting in the driveway, driving 30-40mph down the road, and doing 70mph down the highway.

I have checked all of the grounds, (that I can find.) The battery is brand new and has been tested, and the alternator (although I think, since this happens without the car running, it has nothing to do with the alternator,) is brand new. I have cleaned the battery terminals, as well as the cable clamps. They are both getting a good connection. I have followed both cables as far as I can, and both seem to make good connections everywhere.

I'm just about running out of ideas here!