when checking the battery connections, check both the terminal on the alternator, as well as the #4 cable between the battery and the alternator!!!!!
Earlier this yr, I was fighting a similar problem. The alternator connection looked good, but I cleaned it anyhow, as well as the battery connections. The jeep left me stranded in a parking lot. Once I got it home everything looked OK. But when I was under the Jeep, the RED cable between the alternator and the battery was swollen............. The cable jacket had been kicked, and the copper cable was deteriorating .... Was an easy fix! Was dump that I didn't see that previously!!!