Not Surrender... Not Abandonment... Something else?

[Gunnar F.]

Hi, all. Texas residential lease expires December 31, 2016. Tenant has paid all rent and appears to have vacated the house. Though she has left significant property that appears to be worth a couple thousand dollars. She is totally non-communicative at this point. In her last message she stated she would probably leave in late November but was happy to pay the December rent as owed (again, she did pre-pay this).

Technically, she has neither Abandoned (because she paid her rent) nor Surrendered (because she provided no definitive written notice specifying her date of Surrender) under the contract.

First question... I would like to enter the property because it looks abandoned and we have a neighborhood issues with vagrant break-ins. Can I do so or must I wait until December 31? 

Second question... how long does she have to leave a forwarding address before I can consider myself totally released from any obligation to refund her security deposit?

Appreciate any advice!