Are my scores enough? Or should I retake?

I agree with Dillesh4096, regardless of the two-year gap you more recently mentioned. Since the other parts of your application are in good order and you have demonstrated in the past that you can handle the IR and AWA, there is no need to subject yourself to the GMAT™ for a third time in a rush bid to round out everything. Consider, what would you do if you landed somewhere in the middle, as often happens with back-to-back testing experiences, say, a 690-700 total score, a 5.0 AWA, and a 6 IR? Or what if the score were lopsided in some other way? Would you feel any better with an 8 IR but a 680 total score? I would go ahead and apply. If you were unsuccessful at earning your place in the program, then you could consider a retake.

Best of luck.

- Andrew

Then I will focus on the letter of motivation for both schools and the research subjects presentation.
One more question: at INSEAD it is not required from my side to apply to the program since my last degree courses were in English. However, I am taking the test anyways for HEC Paris, where it is required. If I submit my TOEFL score, can it be an indicator that my level of English is up to their standards despite the AWA score? Or there is no need for me to send unnecessary test results?

If the school does not ask you to submit something, then do not submit it just to make yourself look better. Following instructions often works better. On the other hand, if you contact someone from the program and ask about sending a TOEFL score, and you are told that it will be considered if you submit it, then by all means, have at it.

- Andrew

Thank you!