1. Connect to TIC with TDP and select the automation script to be updated.
2. The connections in that automation script will be loaded automatically.
3. Update the connection and reconnect.
4. Republish the automation script.
(See attachments for screenshots of each step)
After changing the connection information and republishing for one script, that should update all scripts' database connection information for automations published to TIC that have the same script owner and the same connection name.