How do I get 'New Terminal at Folder' functionality when dragging a folder to the terminal dock icon?
I was able to do it two different ways:

First, drag and drop the folder directly into the window you want the path copied to by clicking and dragging the folder, using Cmd+Tab to open the Terminal window you were working in, then dropping it on the Terminal window.

Second, drag and drop the folder onto the Terminal icon in the Dock like you were before, but hold it there for a few seconds until the window you had open appears, then continue dragging the folder into the Terminal window and dropping it there.