Tammy may be completely telling the truth when she says they 'met' at an airport. However, that meeting may not have been unplanned. If Elizabeth knew she didn't want to be a mom, she may have been putting 'feelers' out on the internet, and browsing adoption sites. They may have connected this way and arranged to meet. Tammy would be 'telling the truth' when she was asked this question or told this story, and thus pass a lie detector test. This could be true of many of her statements-true, but not fully or honestly explained. I don't think Tammy personally handled any of this, if this is the case-which would account for no deception on her LDT. But I also bet she put Elizabeth in touch with someone, who directed her to the next person, etc etc. Tammy would not be lying when she said she knows nothing-she probably does know nothing. She may have only opened the gate for Elizabeth to walk through.
Elizabeth could have tossed he phone in someones car through window, or even just left it on a bench. Someone may have picked it up, then discarded it, or seen it in their car and tossed it out. Or she may have headed that way (per pings) to arrange a meeting, where she then transferred the child to a 'couple' as she said. Only not a random couple-a preplanned 'meet us here at this time' couple. . She didn't have all that much money, but may have gone to Miami to wait for a further payment, to an as yet unmade bank account, or some subtle means of delivery.
I do believe he is illegally adopted out, to someone who wasn't too concerned about laws. With the airport(s) in close proximity, he was probably flown out of the country. He could be issued a new birth certificate to his new family, and re enter the country with his new family, or remain where he is, if they are residents of another country. These could be fake papers-but they could also be real, depending on what country and how well the bribe system is and how honorable the government is. Real papers can be obtained, if people want to pay enough and if they have an 'in' with certain agencies.
If it is a 'black market' adoption, I am sure these things happen fairly often, and usually involve a member high enough in agencies to help these children move in the black market.
I do know if you want something bad enough, and have money to pay for it, you can get it. Laws or no laws-there is always someone making a living circumventing these laws, and providing people with what they want for a price.
This is all just my opinion.