You would file as a part time New York resident and have the opportunity to report the income you earned while living there.
You will need to file a part-year return for NY. TX has no state income tax, so you will not file a state return for the time that you lived and worked in TX.
https://ttlc.intuit.com/questions/1901227-how-do-i-file-a-part-year-state-return
https://ttlc.intuit.com/questions/1901560-how-do-i-file-if-i-moved-to-a-different-state-last-year
Q. How do I prove to NYS taxation so that I do not owe taxes to them post August salary?
A. You don't try to prove anything at tax filing time. You wait to see if you get questioned, after filing your part year resident NY return. The most important thing is to get your employer to stop withholding NY tax effective the date of the move. It's when your state return asks for the refund of mistaken withholding that the NYS taxation people wonder why.
Also, take concrete steps to establish your Texas residency: get a Texas driver's license; register to vote in Texas; get Texas plates for your car, etc. If you're subjected to a residency audit, New York will want to see proof that you in fact have become a Texas resident - as evidenced by your actions.
There have been instances where a taxpayer was pursued by their "old" state because they had never changed their driver's license to the new state. So the old state's computer systems "thought" they were still residents of the old state.