I didn't know I used spaces, but you're correct, I do have that checked. If I understand you correctly, I still want it checked, as I don't want full screen applications to be on both monitors.
When I run a full screen application, it fills up my iMac's screen. My other monitor shows my iTunes as before, but the task bar disappears so I can't read the time, and if my dock is there,it disappears as well.
Since I want to know the time, I run an application called SimpleFloatingClock. Before Mavericks, that application always started where I put it on the 2nd monitor. But with Mavericks, when I reboot, it starts off on my iMac's screen, and I have to move it. Also, under Mavericks, if I right click it and then click settings, nothing happens. I really wouldn't need that application (it would be nice though), if the task bar wouldn't dissappear when I run full screen applications.