How do I make sure I am not still connected to the internet when I close my Explorer?

If you aren't loading a page the internet connection will go dormant. It isn't sending/receiving any data, but is able to connect and do so at a moments notice. If you disconnect the connection every time you close the IE, you'll have to wait for it to dial in again. So having it stay connected but dormant isn't really a bad thing. I'm not sure about the Blackjack specifically, but some Smartphones will completely disconnect the internet connection, not just go dormant, if you press the end call key a couple times in a row.