As @trinley mentioned, do not procrastinate on your application. Get it submitted and send the scores that you have. Then, you can submit your new scores as soon as you have them and your application will update and go to the front of the line as an academic admit if you make the minimums.
For your essays, use Word Pad. Word Pad is a lot like Microsoft Word but is just a basic editor and it translates smoothly into the Apply Texas format without hiccups. It is like a notebook online and comes with your computer.
Go to Start
Then All Programs
Then Accessories
If you have it, it should be in there.
If you use MS Word, which you certainly can, when you copy and paste your essay into applyTexas, proof read it carefully as some characters may not translate properly. Proof read it carefully no matter which program you use.
Here are some pointers about the essay https://www.applytexas.org/adappc/html/faq_essay.html
Also try the ACT. Some people do better on it than the SAT. For the ACT you need a composite score of 30 with at least 27 English and 27 Math.
Mays takes only 1000 students and the majority of the seats are taken by top 10% and academic admits. Your chance as an out of state applicant are close to zero if a review applicant. To get a spot, you need to be an academic admit. In addition to the 1360, you need to have the sub scores of 620 Math and 660 Evidence-Based Reading and Writing (EBRW) in conjunction with top 25%.
Mays is usually full by mid November with students who have accepted their offer of admissions. Auto admits who apply late are notified to chose another major.
Best of Luck to you