And Will not go off. Locking the car automatically starts the light blinking. Which seems to be draining the battery.

The owner's manual is particularly unhelpful. I typed in tow-away alarm in the search forums with little success so I hope this isn't an FAQ, but it is a frequently and easily fixed problem.

 

That light always flashes when the car is locked, that is your way of knowing that the security system is working fine. It also tells any would be thief that the car has an alarm.
That is not going to drag the battery down, you'll have to look elsewhere for solutions to that problem.
Look up how to back track on battery drain problems.
Go to page 17 in the PDF of the Owners Manual CLICK to see the description of this feature.

 

Thank you! I've had a bit of gnawing going on on the thermometer so maybe it's a similar issue.