Can I press charges for being falsely accused of molesting a child?

Let it be unless you can prove that the charges were brought with malice.

You should never talk to the police without a lawyer present, that shows that you have no understanding of your constitutional rights, how these things work, or just what can go wrong when you try to handle these police matters yourself. You should never agree to take any test unless they have a court order or submit to a search without a warrant. Unsophisticated and uneducated people can be easily manipulated by the police and others. Highly educated people who have been around more simply say. "I want to talk to my lawyer and I insist that all questioning stop until he is present as those are my constitutional rights. People may think they are innocent and have nothing to hide and then find out that they are an "accompolice" or are infact guilty of some crime. They may be made you inform on others or the police may claim that you made a certain statement that is different than what you actually said. About 80% of people are talked into giving up their constitutional rights, confessing, consenting to searches without a warrant, or making statement that is incriminating. That would not happen if htey had a lawyer to advise them but they are too cheap or unaware to even try to call one, they just listen to the police who lie to them, pressure them, threaten them, or offer them incentives to talk to them. You cannot sue anyone unless it was a malicious prosecution and that is not what happened here.

No one will accept charges of false report on a child. AND, you are not in the clear yet. You could still be charged. They may be working to gather more evidence.

No you can't press charges. And by the way, you were a complete fool to have gone in there to be voluntarily questioned, and without an attorney present! ever heard of the right to remain silent? Your emotional wreck is well deserved for your stupidity.

You can file a report with the local prosecutors office, but it is up to them do decide whether or not to file charges. You do have a right to file a civil action seeking compensation for any harm suffered.