I replaced my starter and something is draining.

Are you saying you have three different problems? Battery runs dead, engine died, starter won't turn off?

For the starter, disconnect the smaller wire and see if it still cranks when you reconnect the battery. If it does, the solenoid is stuck. They will replace it under warranty. Be sure the smaller wire isn't touching the larger one.

All of these could potentially be explained with a charging system problem but that would not be common for the starter to fail to disengage. If it is related, when the starter engages it would be cranking very slowly.