Hey I was actually about to make a similar post on my own issue after installing LED headlights.
In my case, I installed H7 LED bulbs in the low beams, and swapped the factory parking light in the headlight for an LED bulb as well. High beams and fogs were untouched.
Here is a video of my symptoms. Are yours similar?
https://www.youtube.com/watch?v=21uzmrdZk-k
The video explains it best, but I can answer any questions if you have any.
I spoke with my local Porsche/Euro indy today as well when I dropped off my Boxster to get the rear tires swapped.
He was thinking it's a turn signal switch/stalk issue, or possibly the clock spring. He was leaning towards the turn signal switch due to the buzzing noise from the steering column area when my fog lights are activated (as heard in the video). We discussed it possibly being a headlight switch issue, but he noted that my headlights still activate just fine, so it's probably not the headlight switch. He said on some older euro cars the factory fog lights receive power through the turn signal circuit, which could explain why my fog lights flash with the turn signals.
He also thought it could be an issue with the turn signal relay. My rear turn signals work just fine so I don't believe that's the issue but I'm keeping it in mind.
Note, in discussing the issue we mostly ruled out the clock spring because the turn signals are still switching off with the wheel rotation. But I have researched how to replace it still to get a view of how to disassemble the other parts on the steering column as I am guessing the issue is somewhere in there.
So in my mind it's either the turn signal switch/stalk, clock spring, headlight switch, or turn signal relay.
I have been researching the issue and here is what I have found so far.
http://986forum.com/forums/diy-project-guides/47183-turn-signal-switch-replacement.html
https://rennlist.com/forums/996-forum/589406-clock-spring-tab-repair-here-s-how.html
https://www.youtube.com/watch?v=hux78ZoPcAI
Onto the parts I have found. I have not bought anything yet. Here are some screenshots for reference from a parts catalog I have for the 1997-2004 Boxster. Note, on my 2.7L Base I do not have the bottom left switch that goes to the car computer. I believe only the S and 996 models have this.
I am still researching the issue, but I hope any of the information can be helpful for you.