I replaced the pull chain switches on both my ceiling fan and light kit. When I pull the chain on the ceiling fan, it changes speeds but does not turn off. Same thing on my light kit... when I pull the chain, the light does not turn off. I have to turn them off by the light switch.

You did not replace the fan pullchain with the correct replacement. Order the correct replacement from Hunter Fan - Ceiling Fans, Air Purifiers, Humidifiers, Thermostats

Ceiling fan pull chain replacement and repair - Ceiling Fans N More

I'll try that. I got the part from Home Depot.

A lot of them look the same, and connect the same, but are not the same.

My old switch is a WELL TEC E116997. I know the Westinhouse 77020 is not compatible. Does anyone know what model is compatible?

I just searched this site and found the replacement model... thanks!

Sometimes the model numbers are the same but the switch isn't. Order it from Hunter.