2 years ago when I crossed from Costa Rica to Panama I came across an American who had been in an accident and overstayed and was in the same predicament as you. He basically just drove past the Costa Rica exit towards Panama and had zero issues. It was at the time the only border that I recall where they didn't check any paperwork before you actually leave the country. He then just entered Panama as normal. I then rode with him for a few hours before we parted. If the border is set up the same which I'm sure it is you should have no issues just riding through.
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