if the charge are actually false, then yes you could move to have them administratively expunged. However, you say they did not pop up until a "national FBI fingerprint check" was done. Since only you could have left your fingerprints, I don't think some of these things are false.
The first thing you can do is contact FDLE here in Florida and request a complete background/criminal history. Once you have that it should show what the military found. If you know you were never charged in any of those cases, you can file a complaint with FDLE to assist you in the investigation. They are a statewide law enforcement agency. If that does not help you, then you may need to seek an attorney to file motions in the jurisdiction where the records are located. In that motion, you can request the judge to correct the record as it is not you and by compelling the State to prove the conviction is yours. Be patient it may take some time depending if it is a stolen ID case or misfiling by some agency. Good Luck!!