I understand your anger and your frustration with the system and with your ex. However, people do take children out of the country and not return them, so even if she has no reason to think you will do so, her claim would make sense to the people who hear it. You cannot, much as you might wish, engage in self-help as retribution for her perceived contempt. You should indeed seek a court order. Although you could do this pro se, I do not recommend it. What the court may want to know and see could be quite complex. I frequently see people lose in custody matters because they have not followed exact procedure or because of insufficient pleadings or information in the pleadings. You should indeed talk to an attorney about this.
You will probably need a court order which you can provide to Passport Services authorizing you to obtain passports for the children without the other parent's consent.
If you don't want to spend "a lot" of money on an attorney, whatever that might mean, you can either proceed pro se and take your chances or else you can contact some in your area and inquire about their billing policies and arrangements. You may be surprised to find that the cost might not be exorbitant for a matter like this, and the money will most likely be well worth it.
If the outcome is important to you, pick up the phone and consult with an attorney soon.
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