Do the turn signals work when the 3 way switch is in the Stop Light position or do you get the same thing? Are all the turn signals doing the same thing or just the fronts? Does the green light on the turn signal switch flash even though the signals themselves don't?
Easy test first. Turn on headlights, remove bulb from behind green lens on turn signal switch, then try the turn signals. Do they work now?
If not, check the grounds.
The turn signals and head lights are all grounded to the hood/grill.
The hood/grill is grounded to the frame by a ground strap near the harness in the front of the driverside wheel well.
The flasher module is grounded to the mounting studs in the firewall. The firewall insulation can interfere with making a good ground and promote corrosion of the studs by holding moisture.
The 3 way light switch, turn signal switch and flasher module are all connected. A problem in one can cause problems down the line. Since the LED headlights draw less current, a faulty ground in one of the three parts can cause issues down the line. A sketchy ground can appear fine with more current running through it.
A screenshot from the back of the Repair Manual TM -23-5 showing how things are interconnected.