Texas case law defines abandonment as the relinquishment of possession with the intent of terminating ownership but without vesting it in anyone else. The relinquishment must be intentional, voluntary and absolute. Generally, the critical issue in an abandonment case is establishing the owner’s intent. This may be evidenced by acts, conduct and declarations. However, the mere nonuse of the property is insufficient, in and of itself, to establish abandonment.

The lease terminated and the tenant vacated the home. They've told you they left the home, which verifies intent.

I'm not an attorney but I would say you can remove whatever was left behind and dispose of it. You can also charge the tenant with storage and moving fees and deduct those charges from their deposit.

Before disposing of the belongings, I would give them 24 hours to pick the items up but it would have to be an "all or nothing" situation. I wouldn't let them come pick up a couch and leave the other 30 items which is what they would likely do.