I need help. 2016 Ford Fiesta, My driver side headlight comes on after car is off and keys removed. Randomly brightening until battery dies. Newer alternator and battery have been tested (Working). Headlight switch, bulbs, fuses all been replaced, issue persists. All equipment in acr is stock.
Note: no noticeable wiring issues, no fraying or rat eaten wiring.

 

First off I don't have any diagram for your car and here's my assumption (replace SJB with something else if needed):
Your headlamp doesn't have a RH and LH separate fuses.

Fuse Box -> SJB
Light Switch -> SJB
Flash-to-pass activation -> SJB
Autolamp sensor (if equipped) -> SJB
SJB -> LH (low beam / high beam / parking)
SJB -> RH (low beam / high beam / parking)

The fuses, the light switch(es) and autolamp sensor do not have any knowledge nor can they distinguish your LH and RH, they all send signals to SJB (the brain). When you have issue with just the LH, then you can rule out the fuses, the light switches and autolamp sensor.
It's your SJB that has dedicated wire to LH high, LH low, LH parking, RH high, RH low and RH parking and it's the Flash-to-pass, somewhere along the circuit, and splice to your LH high and RH high.

You said driver-side headlamp is ON when engine off key out.
1) Does it along with passenger-side turn ON during driving at night YES or NO?
2) While driving during day time with your headlamp switch set to off (no parking, no auto, simply in OFF position), does your driver-side lamp stay ON?
3) BTW, which exact lamp stays on, high or low or parking? low + parking? Low only? Parking only?
4) Or you mean only way to turn it off is to disconnect battery ground or pull the fuse?

To find out where the short to power is on your, let's say LH low beam only, I'd begin at the connector of your SJB and check for possible damage wiring. Higher chance to have a short to power there than at LH headlamp connector simply because even if your LH low is shorted to your LH high, when engine off key out the 12v feed to high is cut off at your light switch and flash-to-pass switch, and during driving your LH low beam would look brighter than your RH.

 

Referring to your pix,,, why do the headlamp lens appear different ? Is that moisture inside the passenger side lens?