Turn signals are not working?

Okay, do you mean the dash indicators light up IE left and right? Do the marker lights work? I suspect the flasher relay myself. It has two sections in it, one for hazards and one for regular turn signals. I would go to the flasher and check the wiring. Because the hazards are working that section is likely okay. That would be the White wire red stripe. The light blue is the turn feed. Probably the easiest test would be if you reversed those wires and see if the turn signals return. Or just swap out the flasher relay.

Okay, with the dash lights coming on with the key I have a couple more questions, does your truck have DRLs or Auto-Lights? One or both. Do they come on if you turn the headlights on as well? There are a few different wiring diagrams to sort through. The indicators though should only have power when the front TS are getting power and that may indicate something else as that power would need to be coming in from somewhere. You said the bulbs are all good, one or both front TS bulbs, do the dash indicators go out?

Okay, the lack of DRL and auto-lights should make it a bit easier. This does have all stock lights correct? Have seen some of the LED parts cause problems. Attached is the full diagram for the marker and turn lights as well as the headlights. The issue with the dash indicators being on as soon as the key is suggests there is power feeding back into them from somewhere, but none of the circuits should have power in them without the lights being on because the indicators should only be powered when the switch for turn or hazard is on. As much of a pain that it is you might try to narrow down where that is coming from by pulling fuses. Get it so the lights are on then pull Fuse 2, 6, 13, 23, 25, 36 40 and 42 one at a time to see which is feeding voltage back.

Yeah, very familiar with working in the winter. My first shop job was outdoors 90% of the time, either jump starting or towing cars. Back before everyone had fancy roll back trucks!
Not a rush, we will be here.

Okay, fuse 13 powers only the flasher relay. The most likely item would be if the flasher relay has failed as a switch failure causing this type of issue would be very odd but not impossible. To test if it's the switch would be to find the flasher relay, remove it and then apply power to pins 1 and 4 and see if either one lights up the indicators. If it does, then move the switch and see if the lights change. If not, then it's the switch. If however, the switch actually turns on the correct lights, then it's the flasher.

Okay, that could be related to the way the system checks for bad bulbs. Ford doesn't really have that even in the dealer information. However, that shouldn't cause the issue you have.
Did that test light up the exterior lights? IE move stalk to left apply power and left turn lights come on? Turn to right and right comes on? If not, then replace the switch.

Should be the blue wire but you can test with both.

Ignition switch doesn't affect that part of the circuit. It simply turns the power on and off. If applying power to the blue wire let the lights come on but then stopped the power is not getting to the blue wire, I would swap out the signal relay. Ford # 5L3Z13350AA

OOF, just a bit of rust. Looks like a NY vehicle.

Hi there you fine fellows. I like to follow along to learn solutions. Had to add a tidbit. I'm in Wisconsin too. The land of where they throw a pound of salt on an ounce of snow.