When starting my '79 Ford F700 bucket truck, the starter motor continues to crank after the engine starts and the key is released. Is the starter bad, the ignition switch faulty, or the solenoid switch bad?
As Paulster2 says, it's more likely to be the solenoid. To check this, use a multimeter to watch the voltage on the solenoid connector (the smaller of the two on the starter) as it's cranked - it should go to 12v(ish) when the key is turned, and back to 0v when it's released.
They're usually fairly integral to the starter, so it's normally a case of replacing the whole starter, but you might be able to get your existing one refurbished - a lot of places dealing with starters for older vehicles will do so on an exchange basis, so they will send you a refurbished one, then you send them your old one, which they then refurbish for the next person. (I suspect you know that, but other readers might not...)
I have had a few things cause the same problem. There is a relay for the starter on the fender well. Some auto parts stores refer to it as a solenoid. When it gets warn sometimes it will drag a little and not disengage all the way. If that happens you will have electrical problems like gauges in dash acting weird, sometimes you will have problems with charging. Also if the bushings in your starter is shot it will cause it to bind not letting it kick back. If the problem is in your ignition you will be able to tell because when you start the vehicle and then leave of the ignition it should come back a little. If it doesn't then it is sticking. I have had all the things I mentioned above cause the same problem. I would start with the cheapest first if you don't want to take it to a garage.