How do I find out if my ex-wife got remarried and, if so, can I stop the alimony?

You have two parts to your question. The first is how to find out if she got remarried.  If you have an idea of where she got married, this will be much easier.  More and more counties now offer online systems for looking up marriage and divorce information.  Each state also has a central reporting data base for marriage information.  They usually post forms on their websites for requesting this information.  A second option is just to google her name.  Often, the local newspaper will either run wedding announcements or post marriage applications as a routine practice-- then you can followup in that county for a copy of the marriage license and return.  Amazingly, Facebook and other social networking sites are a goldmine of information.  In their moment of joy, people will post pictures of just about everything, including their wedding day.

Once you confirm that she's married, then you can petition the court for a modification of your alimony obligation.  Whether or not they will terminate the obligation will depend on a variety of factors including whether it was court-ordered or contractually obligated.  If you are paying child support because you contractually agreed to it (like in a prenump), and you did not include a remarriage clause, then you will have a harder time getting it undone.  If the alimony was court-ordered because she could not support herself, then you will have a better shot at getting it undone if there is now someone present that can support her.