As a UT Austin researcher, you own the copyright to your creations. As the copyright owner, you have the option of assigning non-exclusive or exclusive rights to that work to third parties, like publishers.
You can retain copyright to your work through the use of an author addendum or through conversations and negotiation with your editor. If you are unable to negotiate to retain full rights to your work, you may still be able to share a copy of it. Most publishers allow the author's final manuscript to be shared publicly. The author's final manuscript is the version of your article that reflects all changes from the peer review process, but that has not yet gone through final copy editing and layout work. It is usually a double-spaced .doc file. If you are curious about policies from a specific journal, you can find more information on Sherpa Romeo. Sherpa Romeo aggregates publisher open access policies and provides a summary of that information on a journal by journal basis.