Last month and security can be asked for in a tenancy at will, but is not required. Now that tenancy has started, though, LL is in a difficult position to ask, although could evict on his theory that you are now at will, if you refuse. However, then I would question whether you are at will if you and LL both signed the lease, unless the lease required all tenat signatures to be effective. Anyway, unless you want this to blow up and be a mess, this is all a matter of negotiation with the LL. Why can't he give a lease now if that was his intent originally?
To questioners from West Virginia & New York: Although I am licensed to practice in your state, I practice on a day-to-day basis in Massachusetts. I answer questions in your state in areas of the law in which I practice, and in which I feel comfortable trying to offer you assistance based on my knowledge of specific statutes in your state and/or general principles applicable in all states. It is always best, however, to work with attorneys and court personnel in your own area to deal with specific problems and factual situations.
I would not put down a last month/security deposit without having a signed lease. Although your landlord is entitled to ask for them, if anything goes wrong and he fails to properly maintain the deposit, it will be more difficult for you to prove the wrongdoing in court without written documentation. If you think that your LL is a trust-worthy guy, and that he honestly isn't trying to trick/screw you over, it might be worth paying him simply to keep the peace. However, the fact that he wants the security deposit and last month rent now, rather than when you initially almost signed the lease, is a bit suspicious.
Given that he wants you to stay and sign a lease in August, I suggest you ask for the lease now. It would be simple for the LL to alter it so it runs from April-August. Once you have the signed lease in hand, then you can write him a check.