You replaced the 1157 bulbs with LEDs, but what? As I'm sure you are aware the 1157 has two incandescent filaments that share a common ground.
Do you know if the 1157 replacement LEDs are bidirectional? This is really two (or more) LEDs that are in parallel but with their polarity reversed so it doesn't matter how they are installed. I can't see why they would be since they are probably in a bayonet style package. You could test for that by connecting 12vdc one way and then reverse the polarity. If it lights in both directions then it is bidirectional. That could begin to explain the screwy operation.
Or...I suppose one or more of the LEDs could have a high reverse current- can you measure mA in both directions?
Bidirectional they are not (just checked it). I even tried putting + to one "filament" and - to the other (not using the case ground at all) and it just ever so slightly lit a few LED's, but it was barely noticeable. Now the strange observation... I get 0.25A when + is on the bright terminal "filament" and - is on the low "filament", and the bulb base starts to get warm. The LED bulb has a resistor on the low and a diode on the high, I wonder if it needs another diode in series with the resistor or is that a purpose built feature? The bulbs I have pull 0.10A on low and 0.30A on high (my taillight LED's pull 0.85A on high and 0.30A on low, but these are very bright and I'm not having issues with those). Hmmm... might be worth a try to install diode's on the low filament... what do you think?