x2....get the battery checked first, 6 years is about all you'll get out of them and could be less depending on the number of times you deep discharged it, amount of load (e.g. accessories), and the climate you drive in.
During that 2 week period, were you driving the truck or was it just sitting there? It's possible the alternator is dying and not able to keep the battery charged up.
In theory the car can run off the alternator alone, but I'm not 100% sure. The battery acts as a capacitor too by helping sustain the 12V level during heavy loads. In other words, the alternator may put out 12V under no load, but without the battery, the loads on the alternator may bring the voltage below 12V and the system may not work as expected.
However, one way I've debugged the system it in the past is as follows: With the truck off, unplug the alternator from the battery, measure the voltage of the battery, start the truck for a few minutes then shut it off. Measure the voltage of the battery again...you'll notice that it is lower than it was before because the alternator was disconnected and not charging the battery. Now, plug the alternator back in, start the truck and measure the battery voltage...if the new voltage is higher, this indicates the alternator is charging up the battery and therefore the alternator is probably ok. Finally, shut the truck off again and measure the battery voltage again...did it go up or is it still down? This indicates that the alternator tried to charge the battery, but the battery did not charge up due to old/bad cells, etc.
If you suspect the battery, take it to get tested.