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Yes, definitely measure both current and voltage. Buy a "amp clamp" or clamp on AC ammeter even if it means spending $100, its a safety things.
A circuit breakers current rating is the HOLD rating, not the trip rating. A 50 A breaker will continuously pass 50A. It may not suddenly "trip" until 250A.
Scary thing is how much over-current they will pass without tripping as fast as one might expect. I once had an automotive 30A breaker tested that didnt "break" quickly until 100 AMP!
Tripping doesnt scare me, heat does. The plug in connection getting hot enough to melt may be all the problem. Heat = bad electrical connections and there is no such thing as "checking" them by looking or disconnecting and reconnecting. Are they burned (if theres melting, definitely). There may be a marginal overload on the inside breaker panel, but not much future messing with it without voltage and current readings.
Breakers dont normally get hot just from failure, and when they do, there should be burning or charring of the plastic body where the terminals connect. Wiggling the terminals might show one breaks off because the body has overheated. Crimp-on or loose screw connectors are the most likely source of failure.
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