Having just went through something similar, only with a oil beep-beep alarm when the engine was off, but the key was on, I found it to be a bad alarm module. It could be that, but it's a costly gamble to replace it. How many temp senders? My 150 has two, one for the alarm and one for the temp gauge. I would say you either have a bad module or a sender wire grounded. Take a multimeter and test for continuity to ground from the sender wire in the harness to the engine block. I believe it should be tan. There's no way a temp alarm should sound with a cold engine.