You are prohibited from possessing a firearm if you are: a) convicted of a family violence assault; or b) convicted of any felony. You don't say whether or not the two assault convictions are for family violence, but the state jail felony prohibits you from possessing a weapon for 5 years after your case was discharged. Under Texas law, it is not an offense for you to possess a firearm after that period. However, federal law prohibits you from possessing a firearm FOR LIFE, and you could be charged accordingly. There are some nuances in this law that some lawyers think allow you to possess certain firearms (for example, black powder arms or firearms that have not traveled in interstate commerce), but I'm not 100% convinced that the federal government might not try to prosecute you if you were found to be in possession of a firearm that met one of those exceptions. The only way I think you could legally go hunting without the possibility of arrest is with a bow.
If you want to explore the exceptions in a little more detail, I'd contact a local criminal defense attorney who is familiar with the Federal firearm law. Good luck.
Disclaimer: This answer is provided as a public service and as a general response to a general question, it is not meant, and should not be relied upon as specific legal advice, nor does it create an attorney-client relationship.
I would agree with Mr. Walcutt.
I would also like to point out, that under Texas law, if you are convicted of a felony you cannot possess a gun in any capacity for 5 years. After 5 years, you can only possess a firearm at a premises at which you live. If a felon possesses a firearm within 5 years or on a premises other than where he/she lives, the offense is a 3rd degree felony punishable by up to 10 years in prison and a $10,000 fine.
As Mr. Walcutt pointed out, the federal law is much harsher.
Robert is absolutely correct about my oversight; in Texas a felon can only have a firearm in their home after the five year period, so hunting would be out under Texas law as well. I don't believe that Texas has the same definition of firearm that allows as much possible wiggle room as Federal law.
Again, bow hunting may be the only hunting you can do legally.