Alarm goes off when I press unlock on Fob.

I would be looking in the service manual for how the alarm system is supposed to work. Then I would be studying wiring diagrams for the system as well.
On a typical alarm system, I would expect the alarm to sound if you "just unlocked" the car- say by having a window open and reached in to pull the inside release handle or pressed the power lock switch. There might be a switch on the exterior lock cylinder that would disarm the system if you used the mechanical key. You would have to read up to see if the manual describes how the remote entry should tell the BCM (?) that the alarm should be disarmed.

It seems that the keyless entry system is unlocking the doors as it should, but the alarm never got the memo that it's all OK.

If someone has Techstream, might want to see if security / alarm system can be turned off?
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I would be looking at the position of your FINGERS when you push that "unlock" button.......to be absolutely SURE that you aren't also pushing on the PANIC button sometimes.

 

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You might try taking apart the FOB to clean the contacts/interior. If you do end up needing to replace the FOB housing ($10), be sure to save the RFID chip before recycling the old FOB housing. Does your second FOB also behave in the same fashion?