T&P valves will only dribble on an over-pressure condition, just enough to to keep the tank pressure below the valve's rated setpoint. When water dumps out as you describe, it's usually because of high temperature.
I wouldn't waste the time to change the T&P, it might be the thermostat failing and the T&P was only doing it's job, fortunately, but only depending on how one views that.
Go for a new tank.--
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