You can also request FBI for a copy of your record.
This answer is of a general nature and should not be relied upon as final, nor is it intended as legal advice. Consult with a qualified attorney before making any legal decisions. Gen Kimura, (832) 247-6932.
If you paid a fine and were released from jail, that probably means you plead guilty or no contest and the judge found you guilty instead of placing you on deferred adjudication. You can contact DPS to get a copy of your criminal history, or try this DPS website - https://records.txdps.state.tx.us/ DpsWebsite/CriminalHistory. I don't know that either of the charges you mention would keep you from getting your nursing license, but you may be able to contact the Texas Board of Nursing at (512) 305-7400 to find out. Of course, you can also contact an attorney to help you with this concern.
This answer is just to provide you with some thoughts on your situation and does not create or establish an attorney-client relationship.
You can write the TDPS at the address below and request a form that will request a background check. It costs $15.00 to get the results. You will need to go to your local law enforcement agency and get a new fingerprint card.
Further, it has been my experience that the nursing board will not restrict you from being a nurse for the minor violations of which you speak.
Texas Department of Public Safety
Crime Records Service
P.O. Box 15999
Austin, TX 78761-5999