If you look at the wiring diagram this is a side effect of putting a diode (LED) in place of a regular light bulb. Normally, when the headlights are on (and I'm assuming now yours are on all the time since the term running lights was thrown out there) with a regular bulb there are two voltages from opposite sides canceling each other out, so they stay dark unless the signal is on. With the diode, only one of those voltages is seen by the diode so they stay lit all the time. My Cutlass does this when the headlights are on after I switched to all LEDs. I know why it does it I'm just too lazy to fix it right now. The only way to fix this would be to use some sort of bidirectional LED (I think they exist) or otherwise modify the turn signal circuit.
As for the LED/solid state flashers, there are two different kinds, one with a ground and one without. In my cutlass, I had to use the grounded one for the hazards (on the fuse block) and the inline for the turn signals (up behind the radio sort of) ... The only way it works. YMMV.