My blower motor will not turn on. When the switch is on there is a buzzing sound but the motor will not turn on. I checked to see if the fan would turn freely and it does. It stopped working right after we had basically overloaded our amperage (60amp service) with A/C, dryer, and hair dryer running together.

No power to the fan motor itself could be a problem.
It could also be the motor capacitor if so equipped.
And it could be the fan motor itself.
And if the fan is not operational in the A/C mode it could be the fan relay.

Many choices so you will need a meter to figure it out properly.