If the debris and belongings have been incinerated, then there is no cause for concern. I do wonder why it has to go all the way to Louisiana from Dallas, though. Is Dallas contracted with that landfill already? Some urban areas ship their trash a ways due to enviro NIMBYs shutting down local landfills.
It was sent to the incinerator(aka Thermal Oxidizer) in Port Arthur which is very close to LA. The upper Texas coast into LA and MS/AL has a significant petrochem industry that generates a lot of this waste so there is probably a lot of this type activity in that area.
Technically speaking this particular waste would be called medical waste which is component of hazardous waste, which is no different from the waste that a dentist or abortion clinic has to pay to be collected, hauled and disposed of.
Lots of hazardous waste gets incinerated at cement kilns because they operate at the same temp as a hazardous waste incinerator. Lots of hazardous and medical waste gets illegally dumped and some of is illegally disposed of in municipal landfills.
There's lots of illegal activity and horror stories, but if a generator pays a hauler who then dumps, the generator is financially liable for clean-up.
There are lots of regulations, paper work, and reporting requirements.