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Not only do you need deal with the regular flasher,but also the 4-way hazard flasher cause it's part of turn signal circuit(I believe this is the case) Do a search here on LED turn signals.
Did you mean to link this one or did you upgrade to 3157s?
I'm wondering if you have one of the bases backwards where it plugs into the trucks wiring. Altho I'm not real sure that would make it behave the way you say it does.
I'd do a process of elimination thing where I would unplug one lamp at a time, note what happened and move to the next. I'd treat the LEDs with the resistors as 2 different connections, meaning I'd pull the LED for the socket, note the difference, plug it back in then unplug the harness from the truck's socket and note the difference.
You could try a LED flasher, most autoparts stores have them in the LED goodie isle. You may have to make a jumper and will have to connect a wire to ground but it's not that hard.