In this case, I wouldn't refer to the sentences as being correct or incorrect, but as being good or bad.
(6) is, by far, the worst of all, because both participial clauses ("my fingers aching" an absolute one) are adjacent to each other. Such juxtaposition is not allowed.
All the others are acceptable, but personally I would prefer full clauses with a subordinating conjunction to express the intended meaning:
- Although my fingers hurt, I kept on writing the draft because I had little time left for submitting/to submit it. (much better than "for its submission")
- As I had little time left for submitting/to submit the draft, I kept on writing it even though my fingers hurt/in spite of/despite my aching fingers.