Let's disect your situation. Your lease is in effect until 7/31; and the lease states that you must give 60 days notice to vacate.
I take it, you would prefer to stay where you at least until sometime in August; and you don't mind paying the full August rent even if you move out a bit earlier.
What I think I would do is just do nothing. I would not sign a new lease, and I would not mention again to the landlord or management company my intention of moving. As I see it, as of August 1, your tenancy automatically becomes month to month. I would pay one month's rent on August 1 and give notice in writing after the rent is accepted if you are prepared to move by August 31.
I would avoid signing a new lease by finding fault with it, or just neglecting to do it. It sounds to me like your landlord is guilty of a certain amount of gamesmanship and you need to step up and do the same.
You could be opening yourself up to some headaches, such as recovering your security deposit, or being charged as a holdover tenant, but, I would take that risk. And no, I don't think that your landlord can charge you until 8/22 and expect you to vacate on 7/31. Remember you have no lease to break after it expires.
Hope this helps.