There are several ways to check your assignment submission status including looking at the Assignment description or to check your Course Timeline.
The first way is to go to the Assignments description. If the assignment is incomplete, you will see a pencil icon.
If you have a green check for the specific assignments, then you have successfully submitted the assignment.
The second way to is to visit your "To-dos" from your course timeline. More details on tracking your progress here.
Under the "To-dos" section, you will notice the status at the right for each assignment in the course, including their respected due dates.