@TheAvidSeeker - if I remember correctly last year the link to submit your housing deposit was not available until 1-2 days after the complete application was submitted.
I believe September would be safe. Most all people I know admitted for Fall 2016 that made a point to submit their app for those priority deadlines- Oct. 15- all got their first choice.
adding info: https://admissions.utexas.edu/apply/housing#fndtn-t48-before-you-apply