If you're truly concerned about your personal life being discovered or risking arrest (no matter how remote), I wouldn't stay in the US. Prostitution is legal in the US only in Nevada and then only women may be hired. There aren't any male brothels. That means you'd have to travel. The US is suspicious of men traveling to southeast Asia for sex tourism and will prosecute men they believe might be engaged in underage sex. That means cross-off southeast Asia.
It is far better to go to Europe where prostitution is completely legal and discreet. This also helps protect your reputation because you won't be subject to blackmail if you hire a man from a reputable and legal escort service. Europe also enforces STD testing (as does Nevada) which is very important. Prostitution is legal in Canada and always has been, but if you believe you might be recognized, then I wouldn't venture up that way.
It's really important to read-up on each country's laws regarding sex workers because they do vary throughout the EU. Of course The Netherlands is the first place people think of and rightly so because the sex trade is so established and well-regulated there. Because of that, I'd consider The Netherlands first followed by Denmark and Germany. You want to hire someone through a service, not someone who isn't legally licensed to be a prostitute. Choose a country, read-up on the laws and etiquette, visit the service websites (everybody speaks English in The Netherlands), and if you do that, you'll have a fun, legal, and safe trip with a minimum of exposure and a maximum of discretion.