Started stays engaged

What is your question?

Went to start my van and battery was dead. Got a jump and van started, drove for approximately 30 minutes. When I shut it off I restarted it a couple of times, worked fine. A couple hours later I went to start it and the battery was dead again. Tried jumping it again, but as soon as the jumper cables were put on the started engaged and began cranking the engine without the key. Was told it was my starter so I changed it, same thing, new battery, same thing starter stays engaged.