Can an intoxicated person under the influence of alcohol/medication press charges against someone?

It is certainly legal for a person under the influence to sign the sort of document you describe, but it's not a very good idea, because a lawyer does not have to be very sharp to undermine the credibility of a witness who was intoxicated. It is helpful that police officers were there to observe the intoxicated condition of the woman in question. The bottom line is that it's not really a question of the legality of the document; it's a question of how useful or useless that document is in any proceeding where the truth of the facts alleged in that document are essential to the prosecution's case. The problem is that the police typically use tactics like what you've described to secure an arrest, even though it's on the basis of bogus information, because they know that once you're in the system it's easy for the prosecution to beat you down. It would be a good idea to lawyer up in advance so that if you're arrested there's someone there to represent you at the arraignment.  At the arraignment all the prosecution has to do is allege facts that if true indicate that a crime is likely to have been committed. You need someone there for you who can stop that whole process before it begins by pointing out to the judge that the prosecution's case is based on the testimony of an intoxicated witness whose credibility is severely undermined by her intoxication and the completely self-serving nature of her written testimony.