Some games have the option of "Fullscreen Windowed" / "Borderless Window". Basically fullscreen without that annoying border.
This enables you to simply pass the cursor over into the next screen while the game is still fullscreen on the other monitor. The game doesn't minimize while you're in the other monitor.http://forums.steampowered.com/forums/showthread.php?t=2675769
Some games do not have this option and so a program to force "fullscreen windowed" is needed. Link above. Simply alt-tab out into the next monitor. The game continues to run fullscreen, unfocused, while your doing whatever in the other monitor. The game won't minimize while you're in the other monitor.
I have two different sized monitors, so I'm not quite sure what you mean by the changing of resolutions. Mine don't change.