Simple solution: pay with a bank check or money order. It is as good as cash to them and now you have your own receipt because the copy you keep shows that it was made out to the landlord, the amount, and the date. Even simpler, if they have accepted your personal checks in the past, then simply give them a check.
Without a lease, you are a 'tenant at will'. This means that if you pay rent monthly, your rental term is a month. They can give you notice of termination prior to the beginning of any month, that they intend to terminate the least at the end of that month.
This answer is provided for informational purposes only. Actual legal advice can only be provided in an office consultation by an attorney licensed in your jurisdiction, with experience in the area of law in which your concern lies.