First of all, never remove either the battery or alternator unless the engine is off. That's is a good way to turn a battery problem into a battery and alternator problem.

Second, when you weld, it's a good idea to unplug the alternator as well as the battery. Stray current finds it's way through the wiring and ends there for some reason.

From the description, my best guess is that you have a bad battery, which can happen just from it sitting around. Easiest thing to do is to temporarily install the battery from another vehicle and see if you get better charging voltage. You'll probably see more than 13.8V. If it's not the battery, then take the alternator into a parts store and have them check it on the machine. It could be a bad diode.