Tenant is behind in rent and has not been seen for quite a while, and they still have property in the apartment. What are the laws in Texas for property abandonment.

What is the procedure for property abandonment?

'Abandonment' doesn't apply to tenants not seen and not paying, and you still have to go through a normal eviction for nonpayment of rent.
http://www.co.denton.tx.us/dept/main.asp?Dept=42&Link=818 is a link with an easy to read summary of steps for one county, and you would contact your county.
The possessions left behind must be stored until eviction is complete, and in some states, a certain period beyond (I didn't research Texas). Just ask about that when you file eviction.

See other post.
Abandoned property laws are local, but are superseded by tenant law, and you are dealing with a tenant.

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Believe me, you want to evict, most likely you are just not catching them at home ( hint, hint)

If you go to court claiming they have just left, you have an entire issue of trying to notify them to get their property, having to move and store it and more.

If you merely go though eviction procedure, let the court nail a notice to the door and evict them, you will get your property back a lot faster, a lot cheaper and without headaches

I had one house in TN that I did not know better and told the judge had just left, well I had to hire movers to move it out, pay for storage for several months, put publication in the legal section. And then I still had to clean and fix the house up.