Most states in the Deep South set the "age of consent" at 16; I'm not sure why it would be a criminal matter for Gaetz to pay a 17-year-old woman's expenses (even if, as Gaetz has suggested, he was just being very "generous" to an "ex-girlfriend") as they traveled through those states. The bigger problem is that the "age of consent" is 18 in Florida, where Gaetz has a substantial documented history of reckless and criminal history resulting in little or no punishment. Considering also his well-known preference for much younger women, It seems reasonable to me that he violated federal law in the course of traveling to and from Florida. Whether that should be a big deal is, I suspect, a matter about which opinions differ greatly, although one would get the impression that, in this country, you'll get few people sympathizing with him in public.
It's odd that this thread seems to suggest two mutually exclusive alternatives. The extortion story does't sound implausible to me, but I don't how it helps Gaetz,
In any event, I'm inclined to believe that this matter ranks very low in any reasonable list of higlly immoral, dangerously irresponsible, and illegal or criminally culpable conduct by Gaetz, although it may be the easiest to proscecute and the most likely to pull him down,