The post office isn't going to run your crimianl record. They are oly going to process your passport information and send it in. However, it is quite possible that the gov't may not issue the passport as you have a warrant or even arrest you upon entering the country. Thus, you need to clear it up. I would not call a bondman but would call an attorney in the jurisdiction that the warrant was issued in. Your attorney can challenge the identity theft issues and maybe get the charge dropped aginst you without appearing or posting bond. This will depend on the jurisdiction and the charge though.
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Answers to AVVO questions are for general purposes only and do not establish an attorney-client relationship. This answer is not a substitute for legal advice and should not be relied upon without a full consultation with an attorney that can fully look at issues involved.
The best answer is to contact a bails bondsman for advice and assistance. The bondsman may be able to work with you to turn yourself in without too many dire consequences. It is quite possible you would be arrested if you went to the post office, but it isn't very likely--postal employees are not charged with enforcing arrest warrants. The information might be sent to law enforcement, however, so you don't want to leave yourself uncovered.