Hello and thank you for using JA.
Most likely the anti theft unit is acting up, if the device is manufacturers install, the unit might have a back up battery and this battery is having a low cell and needed replacement.
My answer is base on the theory of the following, if the antitheft device is activated, you should not able to start the engine, and since you mention that you're able to drive the car with the alarm going off, this words allready tells me the unit is at fault, but just like what I said the unit might have a back up battery and it's acting up, and please don't ask me where is the battery, I just know the ways of anti theft device, but don't exactly know what model is installed to your car.
The other way you can do is (which I don't suggest) remove the whole anti theft device, and if you needed one, have the dealer trouble shoot the unit, or heve them replace it if needed, or you can go into third party alarm installer have them istall a new one and they will by pass the existing one.
Good luck