There is no law which requires a tenant to provide a soon-to-be ex-landlord a copy of the new lease in order to receive a proper rent refund and/or security deposit refund.
You must pay rent only for the actual days you occupy the leased premises. If you give proper 30-day notice and the last day is any day other than the last day of the month, you need pay only prorated rent for the days you actually occupied the premises. For example - if you gave notice on 5/5/12 with a projected move-out date of 6/5/12 and you actually moved out and returned the keys on 6/5/12, you would owe only 5 days' worth of rent.
Technically, the landlord could serve you with a 3-day pay/quit notice on 6/2/12 demanding the entire month's rent. But if you vacate before the 3-days expires, he would not be able to file an unawful detainer action against you in court though you would still owe prorated rent for the days you occupied the premises.
It's a bit unclear what your goal is.
If you wish to terminate a month to month lease, your only obligation under the law is to serve the landlord with a 30 day written notice of termination. You must then pay rent for the entire 30 day period.
However, if you wish to pay LESS than the 30 days rent and move out earlier, then the landlord could ask you for a copy of the new lease in order to consider an accommodation for your benefit.
Frank W. Chen has been licensed to practice law in California since 1988. The information presented here is general in nature and is not intended, nor should be construed, as legal advice for a particular case. This posting does not create any attorney-client relationship with the author. For specific advice about your particular situation, consult with your own attorney.
You are only required to give your 30-day notice, not a copy of your lease for where you are moving to. As far as your landlord is concerned, you may be planning to go sleep in the local park on the 31st day and it would not be your landlord's business.