Q: Accessory / Ignition Power is inconsistent

Once in a while when I get done driving and I turn my key to the off position and take it out of the ignition I find that I have no accessory power. I then put my key back in the ignition, turn it all the way as if to start the car, and the dash lights flicker and dim out to black but nothing else happens. So I take the key all the way out and then reinsert it. Then I turn it to the “on” position and find that I now have accessory power and my car will start just fine. When my car accessories are receiving power, the battery gauge shows a strong charge every time, so I don’t think it’s a battery issue.

About a month ago I was having this problem a couple of times a day, so I replaced the ignition switch, hoping that it would solve the problem. After that, the problem did not occur for about two weeks, but now it has started happening again. While I was in the steering column, all of the ignition wires and the wire connector looked fine as far as I could tell.

Any ideas? Thanks!

My car has 94500 miles.
My car has an automatic transmission.