Remove the fuse, check the sockets with a voltmeter. One should have 12 volts. If neither has volts, then there is some weirdness going on, because the hot side of that fuse is THE SAME connection to the ignition switch as every switched circuit in the fuse box. I can't see how it would be cold while others are hot unless something's broken in the fuse box itself.
Pull the fuse, carefully put a pin or something into one side of the socket, turn the key on, see if it's hot, measure voltage between the pin and ground. Key off, move the pin, and check the other socket the same way.
Make sure the fuses you're pulling are OK. Put an ohmmeter across the fuse, should have zero ohms. If the fuse is open, then we gotta find why they're blowing.
You said you tried the spare fuse, but you haven't said you measured the fuses with a meter. If the spare fuse blew immediately, then something's shorted in the wiring to the jack, or something's stuck in the jack itself.
What I'm saying is I haven't been convinced the fuses are OK when you pull them. Usually a blown fuse shows visibly, but not always.