Start with the entity that originally contracted you and make sure you have something in writing. Then keep going up the chain until you find a person who can (or is willing to) resolve the issue. Be friendly but firm: "thanks for trying to help; who would you recommend I next contact?" That sort of thing.
Accounts Payable answers to the Controller's Office, which answers to the Office of Finance. The buck ultimately stops at the Executive VP for Finance and Administration, who also carries the title of CFO.
Universities are like any large organization, and in some ways they are more like corporations than government entities. The bureaucracy can be confusing even for people who work there, though, as sometimes there are overlapping areas of responsibility.
For example, if your contract was with an academic unit, it might be paid for by a grant, which gets several University entities involved, such as academics, finance, and research administrators. And if the entity that originated the contract did not follow the correct financial procedures, then the process might need to get restarted.
But be persistent: generally in universities there is no culture that has incentives to stiff vendors, unlike the corporate world. Instead, there is a culture that does not necessarily prioritize prompt payments; on the academic side, this can be especially tricky, as a department or college might process the payment request and wash their hands of the transaction: "hey, I did my part, and it is somewhere in the system."
Universities also ted to have a nightmarish set of systems that do not necessarily intersect and communicate well with each other. In many cases individual entities, units, or departments are kind of siloed from each other. For example: I recently had an issue with a graduate student getting paid. HR, Payroll, and the academic unit were not sure where the problem occurred, and it took me a fair number of phone calls and emails to get the situation straightened out.
Full disclosure: I no longer work at UT, but I was an undergraduate student, graduate student, employee, and instructor there for over a decade, so I know the institution well.
Good luck and keep us updated!