You seriously need to get a voltmeter. They are cheap, and will answer a bunch of your questions.
After it runs for a few minutes, turn it off and carefully feel the big red wire coming out of the alternator.
It may be hot.
Feel it back a little ways from the alternator. If its real hot, odds are you have a shorted battery and that is what is sucking all the juice out of your new alt.
An engine will stall due to low battery voltage, and your light flickering indicates low voltage.
But, unless there is a huge load on the system, your alternator should be able to power the truck as long as its running.
If the battery was also stored all winter, and not kept charged, its probably bad. Especially if it was not stored in a heated place.
A discharged battery will freeze and be destroyed internally.
If you have V-belts, make sure that the alt. belt is not slipping. That would cause the whole problem too.
Fasted way to narrow it down, temporally swap with a known good battery and see if the light keeps flickering.
When you are taking the old battery out, look close at the terminals when you take them loose.
Might just be corrosion.