Other things I've tried: go back into the assignment, turn off sharing and assignment. Leave assignment, get back in (and double check console is empty, history is empty, plotting pane is empty, environment is empty), turn sharing and assignment back on.
Went to my "student" account, got rid of the assignment copy I made before and re-started the assignment. Again: environment and plotting pane clean, packages not loaded, BUT code shows in the console and the History pane (the template script is closed; even though it is open in the assignment I am sharing).
I hope you can imagine that as an instructor/teacher this is giving me anxiety. Is this expected behavior? It seems there is nothing I can do here? Create a completely new assignment and make sure I don't run anything?