It depends upon which state you are in, and whether your travels take you to other states.  Can you provide a bit more information?

It seems doubtful that a bust is for lack of license.  Nobody knows that untli you're busted for solicitation or whatever.  I doubt anyone cares.  But sideone is correct in that more infor -- geography, please -- is necessary.

Laws governing escorts/massage parlors, etc., are often a matter of local (city or county) rather than state law.  Regarding the large size cities most ladies visit, there may be city ordinances that are applicable.  In some situations, it may be a county, rather than a city, ordinance, but most large cities are likely to have their own ordinances.  Prostitution violations are a matter of state law, even in Nevada, where state law allows smaller counties the right to legalize/regulate it.
 In general, violations are likely to be at most misdemeanors, carrying a maximum punishment of something like 6 months in jail and a $1,000 fine.  However, first time (and perhaps second or more) offenses may just be a fine, but it is difficult to easily ascertain what the local court's practice may be.  If the local newspaper has a web site and you can access the archives, you could search to see if you can find any articles about arrests and punishments.  Other methods are time-consuming and not very practical:  1) Visit the courthouse; there are often "court watchers" (usually retired folks) whose entertainment is watching what goes on at court (but normally trials, not arraignments, and the latter are where you are more likely to see massage/escort/street walker cases); sit in arraignment court and see if a case with that type of charge is called; go the the clerk's office and ask how you go about viewing the dockets for prostitution/escort arrests; 2)  Find a local attorney who has some experience in the area, but yellow page ads may indicate only criminal defense, not defense of sex crimes.
 Though criminal records have become a lot more computerized and nationalized over the last 20 years, since these are local minor offenses, one city may not know you have been arrested in another city.
 The bottom question, to me, is are you likely to be arrested ONLY for an escort law violation?  I imagine that is unlikely. More likely is that if LE charged a lady with a prostitution violation, while an escort law violation might be added to the original charges, usually the more minor charge will be dropped for a guilty plea to the prostitution charge.

I don't think you need a liscense to date men

just keep your name out of the advertisement section of the paper and you will do ok.

LE can't touch your advertisements on the internet and if you are traveling, you are bound to have lots of offers from just this site alone.....I do.

I think the other posts are correct that this is a local law issue and is more likely a civil (registering and paying local tax) than criminal issue, but I also believe (off the top of my head) that you generally do not have to get a license or register so long as you are not doing a business locally, and I am contrasting locally with across state lines.  In other words, if you have a business selling coffee and go to Nashville to visit local coffee shops and try to sell them coffee, you don't need any license in Nashville or the state of Tennessee because you are doing it on an interstate basis.  You worry about licensing, etc., where you have your business established (usually setting up an office in a more permanent manner).  That help?  (This all comes from the Commerce Clause in the US Constitution and it is an issue that comes up for major businesses all the time.)