Tone,
Rustico is right on with the advice, however, if you are less than 1-month away from closing your escrow you should be able to get the lease negotiated with a contingency such that if you fall out of escrow you only lose your deposit (or maybe even just part of your deposit) on the lease. The deposit is usually equivalent to 1-month rent. Just set the due date for the 1st months rent to be due 1-day after your close of escrow, and write your contingency appropriately. Ideally you would have negotiated at least a 1-week rent-back to avoid storage fees from your moving company, but failing that, all the major moving companies can easily store the contents of your house for a couple of days until your new lease begins at reasonable cost.
New_Renter