No, you are not obligated to pay rent past the expiration of your lease if you already move out by that date.
Your obligations regarding providing notice of vacation should be set forth in your lease agreement. Apparently, there was no such provisions in your lease agreement.
You should not have been forced to sign anything requiring you to pay two additional weeks of rent. However, if you already signed it, then you are likely stuck with complying with it unless you can get it set aside under the legal defense of "duress".
I assume you are going to move out on August 31 as planned. Your options are to either pay the 2 additional weeks rent and move out on August 31 or not pay the additional 2 weeks and still move out on August 31 (and wait to see if the landlord/management company sue you for the two weeks-- at which time, you bring up the duress defense).
The information presented here is general in nature and is not intended, nor should be construed, as legal advice. This posting does not create any attorney-client relationship with the author (who is only admitted to practice law in the State of California). For specific advice about your particular situation, consult your own attorney.
I highly recommend demanding a pre-move out inspection regarding your deposit. If they do not provide one you will then be entitled to your entire deposit. If they do provide one they have to document any "issues" with the place and give you a copy before you move. Just a bit more protection for you as you go through this period of moving...
I do not represent you or anyone out there in cyberspace unless we have a signed (by both parties) working agreement between us. I am only licensed to practice in CA and these are only my impressions related to your matter.