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If the Starter is staying engaged, after the Key is off:Be sure you have all wires on the Starter Relay connected correctly.Make sure the Battery cable has all the connections & the Starter cable is by itself.
Typically it is the Starter Realy that causes the problem you described, but the Ignition Switch Start wire can also cause it, if the Ignition Switch is bad.To check, connect everything & remove the small Ignition Switch Start wire if the Starter hangs up to see if it stops, if not, then replace the Starter Relay with a Name Brand one.
Not sure which type Starter Relay you are using, but the newer type look like this: