Broken ipad power button? Or is this some sort of software problem?
That's not the way it works on my iPad Pro.  Pressing the Power button turns the iPad display off and on. Holding it down for a few seconds brings up the shutdown slider. If you can't turn the display off and on what do you do when screen turns off? Press the Home button? Really only thing you're missing, it sounds, is your ability to turn the screen off when you desire. Or did I miss something? You can turn the screen on with the Home button.
It sounds as if something may be wrong with your button.