Sounds as if your soleniod is sticking , there are several possibilities that cause this to happen, the contacts on you soleniod are worn out and sticking when you start you engine, another is build up of a combination of dirt,oil and crud is causing your solenoid plunger to stick open keeping the drive gear engaged, or the drive gear its self is sticking in the engaged position due the crud build up or a damaged drive gear or possible ring gear. I do not believe the auto start is not causing you starter to stay engage. My opinion is to check your starter first for being the culprit, pull it and do a bench test.