So now that I know what's going on, here is my 2 cents:
1) I don't think you have a bad alternator or a bad battery. To the best of my recollection, the manual does not specify the voltage at idle speed, and calls for loaded tests to be conducted at 2000 RPM. Even at that speed, the voltage has a wide range, and 13.7V sounds acceptable. It would certainly be more than enough if it appeared directly at the headlights...
2) You definitely have high resistance along the headlight circuit. One bulb drops 1.2V (13.7-12.5=1.2), two bulbs drop 2.2V (13.7-11.5=2.2). That works out to around 0.2 Ohm total (because the total headlight current is about 10A).
3) It is not very practical to put your finger on where exactly that high resistance occurs. I would check the following:
-- Corrosion on battery cables (including the negative one and under the insulation)
-- Corrosion on ground connections to both headlights
Clean/repair those as needed.
If that solves the problem (say reduce the drop with both bulbs from 2.2V to 1.0V), great, otherwise:
4) The positive connection to the battery is long and complicated (especially if you have auto-lamps). It is not practical to trace it, and parts may be too expensive to replace, even if you find the culprit. Instead, I would install something called Relay Headlight Harness (9007 version). These are made by many sources and provide a short, direct, and fairly heavy gauge connection to the battery, where the high current flows. It plugs into one of the bulb sockets and uses the voltage there only as a signal to operate a relay, so it doesn't matter how much resistance is in your original circuit (within reason, of course -- it still needs to draw 0.1A or so to operate its relay). Note that these usually do not provide a direct negative connection to the battery, and just hook to the body somewhere. These connections must be clean, tight, and corrosion resistant, and they still rely on good quality negative connection between the battery and the vehicle body.
If you go that route, check reviews before you spend your money - as with anything, some are junk and some are great.
Good Luck!