while the engine is running, the new starter motor try to "start" by it self, without turning the Key. Could it be a starter relay problem?
starter system on 1995 Chevrolet Blazer
It could be a worn ignition switch as well. Most of the time the Solenoid will not trigger by it self, but I have replaced a lot of bad electrical portions of the ignition switch.
Yes, it sounds like you have to check the started relay.