The expression 'don't drag your feet' can refer to anything. A project, walking, reading, washing, etc. I will just correct the English in your question- 'does it restrict to waking'- one should say something like 'does it only apply to walking?' Also- 'we only have 10 minutes left of the project'. Using '...at hand' doesn't really sound right here.

I would say it as "You can't drag your feet! There's only 10 minutes left to hand in our project!"