You can only move spaces which are non-active.
For example, lets say you have spaces 1 and 2. If space 1 is active, you can not move it. You first have to select space 2 then you can move space 1 to a different monitor.
I'm a Spaces user of long-time habit, even though I've always had 2 screens.
I find the best way to get around is to pin specific apps to specific Spaces [then remember where they live, which is the only difficult bit]
If no apps are yet assigned this way, then switch to the chosen Space before launching the app, then right click the app in the Dock & pin to "This Desktop". You can do this without needing to launch first, if it already appears in the Dock, otherwise launch then pin.
Mission Control, as you've discovered, is fine for finding where an app is if you forgot, but not great as a quick way of getting there.
I really tend to use it if I did forget which Space I put an app in, as you can click directly on any of the app's windows in the window 'overview' at the top of the screen & it will switch & bring it to the front…
But… in my ideal world of remembering where each app is, my method of choice is to use
Ctrl ⌃ num to switch directly to my required Space. This is set in System Prefs > Keyboard > Shortcuts > Mission Control [You can only see as many 'Switch' shortcuts as you have Spaces.]
It will remember which app you last had frontmost when you switch Spaces in this way.
Note - if you have an extended keyboard with a 10-key, by default the Ctrl/num shortcuts are set to the 'qwerty' numbers, not the 10-key numbers & you will have to change the shortcuts manually if you prefer the 10-key.
I specifically never use apps in Full-Screen, as they add themselves to a new, random, unnumbered Space & break that key-switching.
Pro Tips
If I ever need, for instance, to add an email attachment from Finder (Space 1) to an email (Space 4) then simply grabbing the header bar of the email & hitting Ctrl ⌃ 1 will move that single email to Space 1, where I can drop in the attachment from Finder.
If you ever need an app to 'follow' you around - sometimes I keep Activity Monitor tucked in a convenient corner - then assigning to All Desktops will make it always appear, whichever Space you're in.
Best Answer
I did this recently to get an iMac to better support an X-Arcade Tankstick and MAME. I do not know that it totally disabled Spaces, but is has effectively eliminated them for me.
System Preferences > Mission Control: Uncheck everything.
Set shortcut for Mission Control to "-"
System Preferences > Keyboard > Shortcuts > Mission control: Uncheck everything
In a Terminal window enter:
For more info you can check out:
Managed Client: How to disable Exposé and Spaces
Mac OS X: Disable Mission Control and Spaces