What battery is it? needs to be a 600cca or larger 660 or 680 if im not mistaken, i have a 850cca in mine. What amp rating is the alternator, should be the 120 amp, not the 70 or 90.
Get the voltmeter in the car or other device that shows you volts so you can watch it as you drive, if they keep dropping no matter what you turn on (headlights etc...) than there is an issue with the carging system. fisrt thing to start with since you have checked the output of the alt and the bat, make sure your ground cables and the main cable from the alternator are in good shape. If there is an issue with them jsut do "the big three" google it, in a nut shell, 4 to 6 Gauge wire added to the main from the alternator to the battery, ground to engine, and ground to body.
Check the output of the alternator with the battery unhooked from it. honestly 13.5 to 13.7 is a bit low for the 98 mk3 tdi. the 3 cars i had all charges at 14.2v for some reason, maybe i just had 3 odd cars. Someone else can chime in on that one. 13.5 is ideal on most cars and my escape is 13.3v so go figure. You could just have 2 defective batteries. what brand, what is the date stamp on it, and CCA?
my car has a drain on both the amp and a few things when i run and its no where near choking out the alternator i have. i run 6 sets of HID's a 2400 watt amp, an 80 amp 12v 3/4 HP dc Air zenith air compressor, electric fans on full blast 100% of the time and a 1200 watt power inverter for various things, i had everything running , air compressor runs for 4 minutes at a time but i had it going non stop for about 20 minutes while blasting my train horns and never had an issue with draining the battery. im sure im putting a hurt on my alternator.
Point is, its not a electrical leak or short or draw doing this, if it was a short, a fuse would blow. Its got to be between the charging system and the battery. Is it the oem alternator?