As you've found, for offline assignments there is no submit button. If you want students to be able to indicate when they are done, you would need to use an online assignment, maybe asking students to add a few comments or a short summary. Alternatively, you could use a Choice activity with just one option, 'Done'.
For tracking a paper exam, you could create a manual grade item in the gradebook, as described in the documentation Grade items. Another possibility would be recording the marks in a spreadsheet, then importing or pasting them into the gradebook, as described in Grade import.