Cool!
So as is often the case, more than one thing wrong, and little to do with the original issue. 
The side lights are brought on when the headlight vacuum switch is pulled to the on (out) position. Note the sidelights will stay on when the vacuum switch is returned to the off (in) position. The sidelight rocker has to be pressed to the momentary off position to turn off the sidelights.
The likely issue is your sidelight microswitch is misaligned, misswired, or needs replacement. If you opt to get a replacement, recommend getting two and replace both the sidelight switch and the headlight switch as they are such a pain to get to and they are at this point a known failure point.
Changing them our requires removing the vacuum switch. IIRC you can do this by loosening all of the dash mounting screws and pulling the dash outward. IIRC the bottom two screws are not captive nuts so you have to reach in with an open end wrench. The sidecscrews also have a Ground for the courtesy/interior lights. You will have to remove the plastic housing from the steering column. Probably easier with the radio out, but mine is a different setup so check with The Veg for that detail. If you are patient and lucky, you will not have to remove the heater control cables, which is pretty difficult.
The biggest issue with disturbing the dash the first time is inadvertently disconnecting various wires by mistake and not being sure where they go. Proceed slowly and methodically, leave lots if time.
The microswitches have a Normally Open (NO), Normally Closed (NC), and Common terminal. The sidelight switch uses the NO and Common as per the PowerPoint diagram.
The microswitches are actuated by the pull handle and can just be misaligned.
You may wish to examine anything else that may need repair while the dash is out.
I also advise adding multi-connectors to allow full dash removal without having to fuss with the multitude of bullet connectors, but this is a mid rather than repair.
I will post a link to the microswitch part numbers shortly. As mentioned, they are readily available but you want the one that is the exact dimensions as stock for the pull to work.
Stu
PS I have been watching the last five minutes of Daytona for an hour and a half. Worse than the Super Bowl clock management! 