Hello Mr. Peterson,
I would try the telephone numbers again tomorrow during the middle of the day. In Florida it is almost 9:00 pm already. (Your question was posted at 7:26 pm Florida time so all of these offices were already closed for today.)
Regarding a paternity test, while you could technically file a Florida Form 12.983(e), Motion for Scientific Paternity Testing, it is up to the judge to decide whether or not he will grant this motion.
The problem is that you were ordered to pay child support many years ago, and the issue of paternity needed to be raised at the time of your divorce when the issue of child custody, visitation, and child support was first raised. To try and raise the issue when the child is 17 years old, it has been many years since these ordered were entered, and you did not comply with the court's order of support (as demonstrated by the $10,000 owed in child support) it is exceedingly unlikely that filing this motion now will do anything but severely irritate the judge - something you do not want to do when you owe $10,000.
Realistically, there is really no chance that a judge will permit a paternity test at this point in time. He will certainly decide that you are requesting it of the court to 1) avoid having to pay the $10,000 and the rest of the support and 2) to hurt and annoy your ex-wife. There is a lot more that the judge can do to collect the $10,000, and you do not want to do anything to anger him into doing it.
So as long as a hearing is already scheduled to address the arrears issue, I would very strongly advise you not to raise any paternity issue. Instead you need to focus on finding a lawyer who can make sure that there is a clear explanation as to why so much money is owed and how you plan on paying it. Remember, the judge is already going to be suspicious and distrustful of you when he sees how much child support is in arrears. Unfortunately, you are already starting off with this bias.
It will not matter if you and your ex-wife discussed paternity issues privately in the past. Unless the court had been repeatedly asked by you over the years because you insisted you were not the biological father you should not raise this issue in court now.
Also, courts often decide that a man is the father of a child and needs to pay child support, have visits with the child, etc. based not on actual biological paternity but on the previous actions of the father - such as acting as the father, supporting the child financially and emotionally while married to the mother. This is true even if the mother already had the child when the couple were married.
Please let me know if you have other questions.