Whenever I try to complete and save either a Character or Setting Sketch template (using “save as”), the work is not saved as a unique file under those titles using either the character or setting name. It simply saves the template itself as completed file. That means I have no way to create additional sketches unless I start from scratch. What could I be doing incorrectly?
The File menu largely operates on the entire project, of which the sketch sheets exist within, as component items within the project. The project is the whole window you see with the binder sidebar along the left. So when you choose File/Save As... you are actually saving that whole thing, the entire binder and everything, to a new spot on the disk. If all you want to do is duplicate individual items within the project so that you can fill them out without disrupting the original, then the Documents menu is what you want. This is what controls the individual component pieces in your project. The Documents/Duplicate/ sub-menu is a great place to start. 
If you are familiar with OneNote you can kind of think of a project in that sense. You’ve got lots of pages and pieces within the pages inside of a project. If you use Save As, it’s like duplicating the whole OneNote project just to make a new page in it, if that makes sense.
Thanks AmberV. Your answer will be very helpful the next time I try this. 
Now, the next part of the problem is: if I have saved it incorrectly, will that affect anything negatively? 
Well, you’ve probably got a whole string of project files on your disk somewhere (wherever you chose to save in the file dialogue). It might be a good idea to go through and make sure you’ve only got one project so as to avoid confusing situations in the future.