i replaced the old starter with a new one but the new starter stays on too. why?
You could have replied to your old post. Keeping the posts together helps us track the progress of the advise.
I advised you to replace the starter because I though I read the starter was just spinning, not engaging the engine. Disconnect the small lead to the solenoid on the starter, and then reconnect the battery. The starter should not be turning. If it is, it must be hooked up incorrectly. If not, then you have a problem sending a 12V signal to the starter incorrectly. This could be a bad ignition switch or a bad starter relay that is separate from the solenoid.
It appears the ignition switch is shorted to the start position since the new starter runs also. If disconnecting the the wiring to the ignition switch clears the trouble then the switch is bad.
Does the starter circuit go through the integration relay, or is the starter relay integral with this? I have seen some freakish electrical problems on quite a few of this generation of Corolla, and all were from the integration relay. I don’t remember how it relates to the starter or the ignition, but it is a very common source of problems in the Corolla.
Unfortunately, because Toyota decided to throw a bunch of relays together, you don’t get to spend $10 to replace the bad one… the bad one causes all sorts of freakish problems and you get to spend $250 on parts and possibly another couple hundred on labor…