In most jurisdictions the state could refile charges against you unless the Nolle was granted with predjudice. The statue of limitation is genrally 1 year on a misdemeanor. I would file to have the cases expunged from my revord.
A nolle prossequi means that the State has dropped the case against you. However, they can refile within the speedy trial time frame. From the day you were arrested on the initial DUI, the State has 90 days to bring your case to trial. However, if at any time during the case, you waived speedy trial, then the State is limited by the Statute of Limitations which is 2 years.
To add to Ms. Jacob's correct answer, you may file for an expungment if you have no other adjudications or withholds of adjudication on your record and remove evidence of the arrest itself.
I agree with Ms. Jacobs. Another thing you need to be mindful of is if speedy trial does run without a waiver, the state could still file within the speedy trial period.If however, the period runs you can seek to expunge this record if you otherwise qualify under statute. Good luck!!