2005 Lincoln Aviator - Lights won't shut off

the lights do not shut off when vehicle is turned off

Sounds like a bad headlight relay.

Is this a new problem?
A new to you vehicle ?

Just wondering if the auto-headlights are turned on, some Ford/Lincoln vehicles have a delay off feature.
The ones on mine stay on for 2 minutes after shutting car off.

I’m not familiar with how the Aviator dash is laid out but on my Lincolns (present and past) the length of delay before they turn off an be adjusted by a switch on the dash. Sometimes the switch is rotated a bit much and it goes past a subtle “click” point and at that point the lights will stay on.
You might go over the light switches and make sure things are where they should be; assuming you want the autolamp feature operative.

If the switches are correct then it could be that the lamps are staying on because of the eye on the sensor is becoming clouded. On my Lincolns the eye is no the top right side of the dash. I assume (right or wrong) the Aviator eye is in roughly the same place. Age and sun fades them out and makes the sensor think that it’s darker than it really is.

I assumed the OP was allowing for that but you could easily be correct. Good catch.