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I am sorry that you had such a bad dealings with child support, unfortunately they will use what ever means they have to try and get support paid including lying. The fact is only Virginia can release the hold on your passport.
You may want to call them and explain the situation. Let them know you paid a lump sum of 5,000 and are willing to start making payments but you need your passport. Often if you are nice and are showing good faith by paying something each month they will release holds in order to get support.
Keep in mind this is just the start........they can suspend any drivers licenses or professional licenses you may hold as well as seize income tax refunds and even have you thrown in jail if you fail to pay the support.
Since there is a issue with paternity you may want to consider hiring a lawyer in Virginia to petition the courts for a paternity based on not being notified until 1998. Based on the number of years that have passed it will not be a easy process and not one I would recommend you go alone on. Since the child is over 18 years of age the courts are not going to be real keen on forcing the child to submit to a DNA test.
In the end either paying it or getting a test is your only option here. The courts are not required to give you a test based on the number of years that have passed. But it is worth a try.
Child support is not in the business of accepting settlement offers. You owe what you owe and they will hound you until they get it. If they are unable to get it now, they will take it from your Social Security or your estate when you pass.
Now, they are often acceptable to knocking off any interest owed (If your order includes interest) if you can pay the full amount within a certain amount of time. (Normally around 1-2 years).
As far as getting them to accept 5K for a 30,000 judgment that is not going to happen. I am sorry, I wish I had better news but child support isn't like other debt you can settle on child support is owed regardless.