This afternoon the YMCA membership coordinator called me back and apologized for the confusion. Apparently, I was misinformed about how membership works. I was given the impression that month-to-month memberships run from the 1st of the month through the last day. If this was the case I would have to pay for two months instead of one to use the YMCA from August 16 - September 16. Luckily, memberships actually starts on the day you pay your fees. So If I paid today for one month of membership it would be good through September 19.
Hopefully, they will talk with the staff about this. With times being tight (financial-wise) most people would not be willing to go to the YMCA if they thought it would cost that much.