When I turn the key on to warm up the coils, if it makes a long motor running sound it won’t start or the dashboard lights turn on then dim. If I don’t hear the sound or they don’t dim the truck will start. This is a year round problem and seems to not be affected by weather, although my wife thinks humidity might have something to do with it. I see it start when it is raining or not though.
I am guessing those “coils” are glow plugs. When cold if they don’t warm up, it is going to be very difficult to start.
It sounds like they are not getting power some times. The noise is likely a fuel pump, you might also need a click sounds for the relay to the coils. The lights going dim likely means the glow plugs are on and drawing a lot of current as they should.
I would look for a relay or the switch. Sorry I don’t know that Ford well enough to be any more specific.