Games not quitting after pressing quit/going into another game

Press start. Go to quit. Go to quick match or server browser at the main menu and find a new game.

I'm not entirely sure what you are asking. No there is not a way to go directly from one server to another without quitting.

Just pressing the Xbox button to bring up the guide does not quit the game. You also have to go to the game, then press the menu button on the controller, then select quit game.

Make sure when you finish playing a game, you fully quit. To do so, you need to go back to the dashboard, highlight the game, press the menu button, and press Quit. If you don't do that and you leave your account signed in but switch to another account, it can start up with where you left off.

If you mean to temporarily stop a game, then press the Start button on the controller, usually. To completely quit out of it, press the white Home button on the controller, go down to the game in the menu, and press Quit and this will shut it down.

You don't have to launch another app or game. Just press the Xbox button to go home or say "Xbox go home", highlight the game in the main window, press the Menu button, select "Quit", and then relaunch the game. If that doesn't work, the restart as Rod
mentions above.