I see it this way:
The SKREEM has two halves, the door lock function (runs on battery power thru fuse 35) and secure start function (runs on switch power thru fuse 37). Either the "door lock" half is acting up (and Josh is wrong, which is rare), or you have two separate issues.

Those two issues are:
1) You screwed up inserting the batteries in the FOB such that: You broke the little tab/bent it so far it does not touch that contacts the TOP battery (has happened a lot before).
OR...
2) You are not syncing the key properly, so it does not sync, hence no remote door locks.
AND...
3) The door lock cylinder just does not work (I mean it's wires to the SKREEM are disconnected/the switches in the look are so corroded, they are open) so the cylinder switches don't tell the SKREEM you are locking/unlocking the door. This has happened before. If you never use the door lock cylinder, you have no way of knowing if it's been "out" for some time.

To verify the door lock cylinder, all ya gotta do is measure the voltage on the "Driver Cylinder Lock Sense (Lock)" and "Driver Cylinder Lock Sense (Unlock)" lines. You can do this in the door or at the SKREEM. Both lines should be at 5 volts or more, until you turn to "lock" or "unlock", at which time, that particular line should go to zero volts. If you NEVER see any 5 volts, the SKREEM is not outputting 5 volts and is the problem. If both stay at 5 volts even when you turn the key, the lock switches are bad/disconnected/etc.

But I'd first make sure the FOB is actually working.

I still think it's the SKREEM.