Is this still an issue for others? I have build out ministering over the last few Sundays.....
Are you using the "Proposed Assignments" tool, or trying to make the changes directly in the "Current Assignments"? One approach might be to make just a couple changes, then log out and log back in and see if the changes remain. If you're using the "Proposed Assignments" then you shouldn't need to worry about leaving things 'in-progress', since no members (besides the Presidency) will see those incomplete ministering assignments until after they are reviewed and published.
And if using the "Proposed Assignments", you could try making some changes during the week, avoiding the potential system overload issues you suspect might be occurring, and also not having to worry about not making all the assignments at the same time.