Healight motor module is fried. Usually it isnt "cooked" but there are key spots on the circuit boards inside them that de-solder themselves over time, due to the voltage they deal with.
There is a thread on here on how to fix them, I will try to look for it, and post the link here. The headlamp module is located on the top of your dash, directly under the plastic pop-up cover that covers most of the top of the dash. It is held in by tension pins. Just carefully pry up on it, starting at the corners and it should pop off. Then looking through the windshield you will see a oval-ish black box smack in the center of the dash. That is the module that controls your headlamp motors.
That is usually what causes them to stay up or not open.
I also find that sometimes if the car sits for a long period of time the module gets "lazy" and the headlights won't open. So I will manually open them (using the access panel under the hood) and then when I try to turn the lights off and on the module will "wake up" and then the pop up lights will work. Sometimes it also takes a couple days for them to get back into the swing of things, but your module could be no good. If you need a module I have one in a parts bin. Good luck.
Out of the eight modules from 87-89 year daytona's I have bad, I have fixed all of them by simply re-flowing the solder joints for the relays. With time, age, amperage and shitty manufacturing the solder joints turn into what is called a cold solder joint which will work intermittant or not at all.
Easy fix if you have an hour and can handle some delicate prying work and somewhat handy with a soldering iron.
87-89 Daytona's have the module in the center of the dash, pry the dash cover up and out and there is a black odd shaped module in the center speaker hole, two phillips screws and it pulls out and disconnect the connector.
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