I post this after searching and reading all of the relevant threads. I didn't really see any situation that matched mine exactly.
Doors lock and unlock with fob.
Doors lock and unlock with button on door if the door is closed.
Doors do not lock by pushing in the handle on either side. The handle pops back out when I push it in and I don't hear the double beep, I have listened carefully but I have very bad hearing. Same on both sides. Nothing happens if the door is locked and I push the little indent on the handle. I don't hear the double beep, I have listened carefully but I have very bad hearing.
Trunk shuts fine and locks / unlocks with fob and button. I have not seen the trunk open indicator on the dash.
One of my fobs started the car but wouldn't respond to any of the buttons. Replaced the battery and now all of the buttons work. Re-tested all of the above with the fob with the new battery, no change in conditions.
So my questions are: Is there a thread I missed that has my conditions with a solution or can someone suggest some troubleshooting to narrow down the solution.
Thank you.

 

Sounds like you don't have the keyless door entry option. I don't have it either so pressing the indent in the door handle does nothing. And the handle pops right back at me when I push it in.

The keyless option has a physical button where the indent is on the handle. Can you check?

 

Thank you. No physical button. I guess that means that it is working as expected.

 

Yes, "Old School" locking!