Left turn signal marker on gauge stays on

do you have a turn lamp burned out?
the year/make/model and pics of the style of your headlight/grille may be necessary...

assuming you have the common style of a headlight assembly with a marker light/reflector underneath, there is a turn bulb in the marker and another in the headlight assembly...
i think the dash light will stay lit if just a marker bulb is out... it will 'click' faster too when you signal
obviously the rear turn bulb could be out as well...
sounds like you may have to inspect your wires for a bigger issue though

theres multiple light styles that might be different ... i know some gmcs and oldsmobiles have the markers and lights combined into one unit

i havent messed with leds on this truck but that may be your issue... leds are not exactly plug and play on turn signals...
no marker light will illuminate a dash light and click fast when the turn signal is applied.
my one marker bulb is pulled out now and theres a dash light illuminated constantly and fast click.
pretty sure your problem is led-upgrade related...