Well, first remove the bulbs from both front blinker sockets. Then, when you turn on the headlights, see if the indicator stays on.
If not, then the power is crossfeeding through those bulbs.
Its faintly possible that the filaments inside have shorted together. Swapping the bulbs from side to side would show if the problem followed a bulb.
Or just replace the bulbs to be sure.
Also, examine the inside of the socket carefully. If the side contact looks funky, its probably the problem. Also examine the ground for that side headlight and the blinker socket.
You can also attach a long wire to a known good ground, I recommend the negative post of the battery.
Then, with the lights on, touch it to the ground part of both the head light and the blinker bulb, one at a time.
If the problem goes away when you touch one or the other, then you have a bad ground there, no matter how good it looks like.
Of course, do the ground testing after all the bulbs are back in place.