When I view the "Drag & Drop" tab it is greyed out and it is not selected. Does this account for the behavior I'm seeing? Do I need to use the "installer"??
Yes, you need to use the installer to install the shell extension.
If you are not using the extension, the "overwrite dialog" is opened by Windows Explorer, not WinSCP. I'm afraid that I cannot control the box opened by other application. However I'll check it.
If you use the extension, WinSCP will be able to handle the overwritting by itself.
If you do not want to install the extension, you will need to drag file only within WinSCP. It will also solve your problem.