The Temp gauge if it is an original type has no electrical connections. If it is dripping? What is dripping? Coolant? If it is dripping coolant then yes it is SHOT, actually if it is dripping anything it is shot. If it grounded out somehow , then I would check the light bulb wiring to the gauge, MAYBE it is grounding to the capillary tubing....

Frank
1948 5 Window Chevy 3100 an absolute rescue that is now back on the road
1953 Buick Super 1 family owned since new
1956 3200 Big Window
1985.5 Mustang SVO
1986 Mustang SVO