Hi Mary - here are a couple of additional thoughts, although I think Cindi's web source probably has your best answers.
First, I would stop making any further payments to the landlord. If he/she hasn't already "spent" your deposit, you should demand/negotiate that both the deposit, and future payments go into a neutral depository - similar to an escrow.
Next, pretty much any agent can get info on the title of the property, to see if the owner already has had a notice of default filed against the property. That may give you some clues as to how much time you have.
Third, I would NOT allow the owner to have a lockbox installed. You should not have to be making the place accessible for "prospective" looki-loos.
Both the first idea, and the third one, should help you negotiate with the owner.
Bear in mind I am not an attorney - I'm just offering a few opinions.