I have a 94 Explorer headlights will not go out even when ignition is off and key out.

By 1994 a number of vehicles were adopting computer control on some of the body functions and it is necessary to be aware of how much control there is over switching, etc. of your car. In order to know these things it is necessary to look at the circuit diagram.

Clearly the headlamp relay(s) are remaining closed or if it has one the electronic or computerised switch is remaining switched on.

In the case of a relay it might simply be stuck or it might be a fault is causing the switch or relay to remain energised and in this case there is little hope of a successful diagnosis and cure until you know how the circuit works, where the influences are and the wire colours to look for to test/disconnect.

First find a wiring diagram...

Good luck

Headlamp Delay System - Autolamp, Aerostar and Explorer
The autolamp system provides light sensitive automatic on-off control of the exterior lamps normally controlled by the headlamp switch . The autolamp system keeps the lamps on for a preselected period of time after the ignition switch (11572) is turned off. The preselected time lapse can be adjusted by the operator up to three minutes.
On Aerostar, the system consists of a light sensor amplifier (13A018) , automatic headlamp dimmer relay (13A025) , and a time-delay headlamp dimmer sensor control (13C105) which includes an ON/OFF switch.
The Explorer consists of an ON/OFF switch, a light sensor and a time delay control which are part of the inside rear view mirror (17700) and two automatic headlamp dimmer relays .
Connections to the vehicle lamps parallel the connections to headlamp switch requiring the headlamp switch to be turned to the OFF position for the automatic control feature.
In normal operation, the system will turn the exterior lamps on when the natural outside light available to the light sensor amplifier decreases below a predetermined level, such as at sunset. Also, the lamps will automatically turn off when the level of outside light exceeds a predetermined level during daylight hours.
On Aerostar, the light sensor amplifier is mounted under a group of slots in the upper instrument panel in front of the driver. The light sensor amplifier must always be exposed to outside light to function properly. Do not place any items over the group of slots in the upper instrument panel which may block light to the light sensor amplifier.
On Explorer, the light sensor amplifier is on the back side of the interior rear view mirror. Do not place any items on the mirror that will block the sensor.
The headlamp dimmer sensor control thumbwheel is located next to the headlamp switch for Aerostar and on the interior rear view mirror for Explorer. The autolamp system is functional for automatic operation when the on-off control is moved from the OFF position. The closer the control is moved toward the max-delay, the longer the lamps will stay on after the ignition switch is turned off.
Pulling out the headlamp switch will override the autolamp. Automatic operation will not be possible until the headlamp switch is turned off.
Can you operate the headlamps manually, do they work ok? I don't like auto-lamps. Just my opinion.
I thought the above info might help?