1969 Dodge Dart - Car in accessory mode without key in ignition

OK. Thanks.
You are correct.
Dome, parking, headlights, and brakelights are always hot.
Wiper, heater fan, radio are on "accessory." which should be hot only when the key is in run or accessory.

It looks to me like the engine compartment wiring harness is a replacement. I say that because the fusible link wire doesn't look like the original from my '67 etc.
This may not be relevant, but its possible not all the wires match up in the right spot.
Another possiblity is the ignition switch and connector has been changed. Do a search here for '67, 68 and 69 ignition switches. There are differences and I don't know if anyone ever posted up if there are terminal position changes, but it seemed like there might be.

Get yourself a 1969 Dodge Service Manual.
or, if you can't, a 1969 Plymouth Service Manual.
Those will have the most accurate wiring diagrams, including the connectors and cavity positions.

You may want to simplify the scheme to focus on the main circuits.
Basically it looks something like this.
upload_2020-1-15_11-17-1.png

Add or modify details to match what you find vs. what's in the '69 FSM.
Here's one we were working on for a guy with a '68 or '69, I forget.
upload_2020-1-15_11-22-45.png