An amp meter I would not recommend but an amp clamp that is hooked up to a scope would be best. Odds are the blower motor is going bad causing a high amp draw to spin it. As for the blower motor resistor use a ohm meter and check resistance on each leg of the circuits. Low speed should have the most resistance and high speed should have the lowest.
As for the melting of the connector that is a result of high current (amps) flow. And before anyone wishes to flame me yes resistance can melt stuff as well but with a new connector and resistor makes it doubtful. The blower motors on most cars are on a 30 amp circuit but typically see no more than 20 amps when blower is running but spikes to 28 +amps on start up on high speed.