You can file a request with the District Attorney Child Support Enforcement regarding the matter. You will probably have to request an Order Shortening Time or the hearing will probably take quite a bit of time. I would expect that you are going to be required to pay the arrears to get the release for the Passport. This will probably require an attorney because it can be difficult to navigate through the child support matters regarding this issue.
My colleague is correct. You have to file a motion with the DA in the state that has your case to release your passport. Most DA's are hard ass about this, because they hear promises to pay all the time that never come true. So you will likely either have to pay off all your arrears, or a substantial portion. It can take weeks to months to get a motion heard. If you are only 5 months behind, you should do everything you can to get caught up, including taking a loan if necessary.
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If all else fails, it may be possible to resolve this with a Chapter 13 which pays the arrears in full. However, if you're only talking about 5 months, this may be an expensive way to shoot a fly with a cannon . . .
The best way to "fix" your problem is to either hire a good child support attorney in Florida to help you, or you can contact the Florida child support enforcement office and find out exactly what is going on and what it will take for you to obtain a passport. Honesty is always best so all you need to do is tell them that you need the passport for your new job so that you can satisfy you child support obligation. All they really want is to be able to collect whatever they can from you to pass on to the other parent. Once you satisfy them that you are genuinely interested in fulfilling your child support responsibility, you should have no problem obtaining your passport. Good luck.