I have, as mentioned, a rather personally fraught relationship with online dating. Catching too many of my supposedly monogamous partners using personal ads to cheat on me left me pretty thoroughly unable to commit to the process. And, when it comes down to it, you have to commit to the process: you are saying, in effect, that you wish to meet potential partners through a service we've all paid for in order to meet other potential partners. You have to accept that it's a perfectly acceptable way to meet someone, and to set down and just let go of your hang-ups about it.
The person with whom you have decided to have a monogamous relationship owes you fidelity, as you owe that person the fidelity you also promised. If that person breaks that promise, I can guarantee one thing: he or she made the choice to do so.