I have no access to your application so I am not in a position to tell you what it states in relation to the second deposit. If it does not characterize that deposit as being anything but a payment toward your first month's rent, then it's not a non-refundable holding deposit.
That said, depending on your anticipated move-in date, your landlord may be entitled to proper notice to quit even before you move in. That is, let's say you were to move in on May 1, and you gave notice today that you were not going to do so. For a month-to-month tenancy, your state requires at least twenty days notice to quit before the end of the current rental period. Unless there's something in the contract that you signed that makes the formalization of the lease contingent upon the execution of another document, or some other contingency that was not fulfilled, your landlord would be able to argue that your tenancy started on May 1 and that your notice given after April 10 would be effective at the end of May.
Of course, if the contract you signed says that there is no lease agreement until you are approved and a lease is signed, then the landlord would not be able to assert that he was entitled to notice. Also, your landlord has a duty to mitigate by seeking a replacement tenant -- although with larger complexes they can make a unit abandoned by the tenant the last they attempt to fill as applicants seek units within the complex.

