If there is no clicking sound when the indicators or hazards are on, that should be the issue. Take the old one out and head to an auto parts store and grab a new one.
The signs definitely point to an alarm/immobiliser issue. Normally, the indicators wouldn't flash unless that was the issue. Does the remote(if it has one) do anything when pressed?
That is what I was asking.
Sounds like it might be a dead actuator. Depending on how confident you are, they are fairly easy to access by pulling the door card off. From there, you can easily diagnose it with a multimeter.