Erratic voltage reading

88 XJ Limited- The volt meter jumps from around 13 down into the the red and then back up to 13. The lights dim and the heater fan wont blow when this happens. It will drop in the red and stay there if I am accelerating. Decelerating will bring it back up to normal most of the time. I took the alternator and battery to Autozone and they both tested out fine. I replaced both battery cables, the positive cable had about a 3 inch bare spot, but it made no difference. Any help would be greatly appreciated.

its your alternator.....when we test them it shows the positive "no load" out put.......its not at all uncommon for alternators to sporadically fail under a load....if I had to bet trust me it would be your alternator

also make sure the braided ground strap at the firewall is tight at the head and the firewall too.

Ground strap was good. The plug in connection from the alternator to the wiring harness was missing the rubber gasket leaving a loose connection. Easiest repair ever, too bad it took 9 months to figure out.