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Check the plug on the back of the ignition switch. It is common to have a burnt up connector on the plug on cars with a/c due to the higher current load. Back in the day when you bought a new Motorcraft / Ford ignition switch it came with a new pigtail to make a proper repair.

Sounds like something going on at the fuse box, as radio, backup lights, and blower motor/AC are powered by ACC and fused by 2 fuses. Might be a bad ignition switch or bad/nearly broken wire for ACC at fuse box. Usually, it is the BAT line that breaks, not the ACC line. Something is going on that when more power is pulled through those lines, continuity is lost due to higher resistance. I'd check your fuse clips first.

While the engine is running, the only thing the starter solenoid does is provide a connection point between the main harness, battery, and alternator via the large post on the side. I don't think the starter solenoid is your problem, unless the connection is loose.

Thanks for the replies! The car is a 69. While I will check what you all have mentioned, I have another theory that I haven’t tested: When I installed the new starter solenoid, it was a black color versus the gray color of the old one. Since that’s when my trouble started, I looked again at the online descriptions of the starters. Apparently the gray one is heavy duty. Could this be the issue? By the way, we made it to Dearborn from Atlanta! The power issue cycles through about three times and then stays on, so we’ve survived the travel. I’m posting on Instagram if you want to see our pics: @stevestang1969

The 8.8 rear end is an entirely different design than the vintage Ford rear that came with first generation Mustangs. I believe the 8.8 was introduced sometime in the late 70's or early 80's. It is not at all compatible with the earlier 8" or 9" design.

Remind us, is yours a 69 or 70? Fusible links started on the 70s, but the only one I'm aware of is at the solenoid, and would cut the main power. Once it is gone, power wouldn't be restored without replacing the link. This sounds like a loose connection somewhere.

Went to wallmart this morning ,didnt want to park anywhere where it may get door dinged so i parked next to a black hell cat and a RT that was same color as mine but with black stripes ,we all three had the same wheels ,i think this was the designated Challenger only parking space ,forgot my phone so i couldnt get a pic!

Mucho beuno ! I like these wheels ! I didnt know it was independent rear end . The car automatically resets the tire pressure sensors as well