Hood ajar sensor wiring diagram?

I couldn't find the hood switch on the original Ford diagrams, but it's here on an aftermarket diagram. The wire isn't shorted, (grounded). That's what it's supposed to be when the hood is closed, so a grounded wire would make the warning light turn off. What would cause this is a break in the wire, (or a defective switch, but you replaced that already). To be safe, I added the drawing showing the location of the hood switch. It's right behind the front edge of the driver's head light.

If the warning light turns off when you push the switch by hand, that switch may be one of the adjustable styles and is out of adjustment. If you post a photo of it, I might recognize how to adjust it.

The next step is to unplug the connector from the switch, then ground the blue / orange wire. If that makes the light go off, the ground wire likely has a break in it. Let me know what you find up to this point, then we'll figure out where to go next.

If you pull of that rubber accordion boot, you'll see the plunger. If you already pressed it by hand, adjustment is likely not the issue. For the adjustable type, when they get out of adjustment, the hood doesn't push them far enough to turn them off, (or on). To explain, here's a photo of a sliding door switch from an early '90s Caravan. See those ribs on it? The first time the door is closed after that switch is installed, it clicks in as far as necessary, then it's adjusted. If you really slam the door, it's possible for it to over-adjust, then the common symptom was the "Door Ajar" light would flicker on when going around hard left-hand turns. It has to be removed to reset it. It was pretty simple to tap it into a deep socket to rest it, then just reinstall it again.

If your switch is adjustable, the design doesn't allow for it to be tapped into the reset position. You would have to pull the plunger out by hand. My thought is yours is either non-adjustable, or, if it is, the rubber boot might be limiting how far you can press it by hand. Regardless, from what I read on circuit operation, your switch turns on when pressed, meaning grounding the blue / orange wire should turn the warning light off. If it does, we have to look at the switch and the ground wire and ground connection.

I'm freezing in sub-zero weather too

Dandy news. Very happy to hear you solved it. Let me know if you have additional questions, otherwise, please come back to see us again.