Just take the alt out and run it to autozone. Takes all of about 15 mins tops to pull and the same to install.
Sounds to me more like you have a bad battery. If the batt has a bad cell it may charge just enough to start the motor but cause other issues. I have seen bad batteries do some crazy crap ...specially in ford's. Wipers turning on and off, blinkers, etc ...nuts.
'01 Silverado ECSB - 5.3L, K&N, BONE STOCK(For now) 7.5'' Fabtech Lift on 20'' KMC XD's wrapped in 33'' Micky's
'72 C-10 SBC - 5.3L, 4L60E, MSD, K&N - still a work in progress.