I am a residential landlord in the state of Texas. I recently had a applicant and her husband complete and sign a rental application, they were approved by me. They gave a security and pet deposit 2 weeks prior to moving in along with a verbal statement that they wanted to move in and were flexible on the date.  They did not sign a lease and on the day of move in they sent me written notification via email that they changed their mind and were not moving in.  Can I legally keep all or part of the security deposit for damages. Unless of course I find a new tenant immediately. 

No, you can't keep any of it. You were remiss not getting a signature on a contract, and the first month's rent.

Can I legally keep all or part of the security deposit for damages. Unless of course I find a new tenant immediately. Hello S:

Yes, you can.

excon

Yes, you can keep it or part of it by law, but the TX code indicates conditions where there is a lease, and you don't have a lease. So I'm not sure who is right.

Does the rental application say anything about deposits? Was the approval in writing? Was a movein date agreed upon in writing so you can prove they planned to move in?

If you can prove they had agreed to move in, they owe at least one month's rent. That would be deductable from the security deposit. If you can find someone to move in before the month is up, you can pro rate what you take out of the security deposit. Once they agreed to move in, especially since they provided money, that sealed the contract.