Lots of issues there and good things to think about.
The answer, like any good legal answer, will be "it depends." It's going to matter on each state's laws. For example, if we reverse it, and make it a non-TX LLC operating in TX, we would look at Chapter 9 of the Tex. Business Organizations Code. The law gives a "foreign entity" the same protections, obligations, and rights as a TX entity but no greater. This is not a big deal since the TX protections are great. So, we'd need to see what the statutes say on each state to see.
It will also depend on the contract between the management company and the tenant. Did tenant sign only with management company? Was management company a third party or also controlled by John Doe? Did management company sign as agent of owner? Does management company have to indemnify owner? Did tenant acknowledge that owner doesn't maintain the property, only management company in the agreement? etc... What are the premises liability laws of RE's jurisdiction? Lots of variables to manage.
You're right in that the law of the "other state" is what will typically determine liability for the damage; however, it's pretty common that the "internal affairs" of an entity are typically governed by the law of the formation state. TX codified it's version of the internal affairs doctrine in Title 3 for LLCs. Each state will have their own statute and/or case law showing whether they do or do not adopt the internal affairs doctrine. What's less clear is what happens in a piercing the veil scenario
I like to do an ownership trick that gives at least an argument to make creditors trying to foreclose charging orders potentially subject to internal affairs doctrine. Also, remember that charging order protection is typically an outside protection; i.e. it keeps your LLC assets in your LLC if you are sued for something outside of the LLC. Texas allows charging orders like most (maybe all? don't quote me on all) states, but Texas does NOT allow a creditor to foreclose a charging order.
And then the biggie: can a foreign state force liquidate your single-member TX LLC? My argument would be no due to lack of jurisdiction. My follow up argument if I lost that would be the "other" state only has jurisdiction over the property in the "other" state. Whether that state can or cannot is a huge can of worms, conflict of laws, how much of a fight put up, is removal to federal court an option, etc..
And unfortunately, even this post isn't exhaustive. Lots of angles I haven't discussed. But, having a strong protection LLC gives you the argument and the chance despite the "other" state's laws.