I'm trying to figure out how I can get the program to cancel the current snap mode when I select another or to default to a "no snap mode" after every selection. In other words, I'd like to set it for a one-time snap.

If you want to override your esnap settings, with a specific one, any one, just use the shft key + right mouse button. It'll show you a floating menu where you can choose a esnap for use once in a specific situation. Then, your esnaps settings will work as usual.
If you want to cancel your esnap settings, you can go to the bottom right of the screen and double click the ESNAP box. It'll set the esnaps off, and if you turn'em on again, your set will remain.
You can also, before picking a point with the mouse, use the NON esnap. It'll turn off the esnaps just for that pick, and then, the set will work as you set it.
Well, I hope this helps...