So wiring wise the two circuits have nothing to do with each other.
When you turn on the headlight switch you switch 2 completely diffent circuits : headlights and marker/dash
my thought is that one of the led lamps is bridging both the marker light pin and the turn signal pin inside the socket . So it's lighting both circuits inside the lamp when you flip up headlight switch shunting your signals .
Again this is a guess ... I would change the front and rear sig lamps back to conventional lamps . Make sure the sigs work again , then change back to LEDs one by one testing after every lamp install to find the offender.
I have seen led replacement lamps that had big u-shaped globs instead of dots for contacts on the bottom causing issues.....
Again my
I dont know what products you used