The statute of limitations only applies if you committed an offense but were not charged. If you were ticketed and failed to appear or take care of them, a warrant can still be active for you. Sometimes doing the driver safety course is an option on old tickets, but most city prosecutors seem to take the position that you have two options on tickets that are several years old: you can either plea guilty and pay them off or you can have a trial to determine whether or not you are guilty of the traffic offenses.
Setting these tickets for trial might be a good gamble; it's been several years and the officers may no longer be with the department(s).
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.
In Houston, and maybe there in Ft Worth, you can call the city hotline and give them various information to use in trying to see if they still have the tickets pending = like your driver's license number, your full name, your DOB. Sometimes when tickets get really old, especially after they have had a couple of those sweeps to pick up offenders, they will just get rid of them - especially if they were before the more organized era of computers.
I agree with the suggestion to set the case for trial if you must. However, you will likely have to post a bond first because there will be a warrant if you failed to appear for court. . . Sometimes it is better to pay the fee to a traffic court lawyer & let them take care of it as they are in traffic court all the time and know the special ins and outs.
Just because a ticket is old, it does not mean that it will automatically be dismissed. The Statute of Limitations in which to file a Complaint in Texas is two years from the date of the offense. However, the calculation of the two year time period may be affected by various circumstances. If your ticket is for an offense that happened more than 3 years ago, then your driving record will not be affected unless you are dealing with the offense of not having insurance.