2003 Buick Century Right Turn Signal Stuck when dark out, car dies when left gets used

Sounds very like an earthing problem. Check the earth on your lights, lh side obviously is doing something funny to the ECU.

Often trailers do these odd light swapping games if earth is uncertain.

Sounds like light sensor grounded out when it turns lights on sending electric to CPU on a ground or your to your dash board .I'd say it's grounded to your turn singnal wire?when two hots trying go through CPU automatic shut down. Try in hooking sensor that turns lights on .most vehicle you can mash emergency pedal down one noch or pull brake up one turn lights off.hope you have luck finding it .them light sensor do go bad like that or it's got warm I'd say kinda melted into a wire .

I have a Malibu 03 with similar tail light problems check for moisture in the tail light assembly and replace entire assembly as the contacts corrode and cause all sorts of problems! Once replaced if there was moisture then check fuse holder for heat damage as sometimes a person will put to large a fuse in to compensate for frequently blown fuses!

Typical of a faulty ignition switch. The contacts internal to the he switch get worn over time. The extra load of lights and the blower motor just exaggerated the problem. Check the voltage at the engine control module fuse or fuses with the lights on. Also check the voltage at all fuses when the car has stalled out