Addison is actually a safe place overall. As Addison caters to the more affluent crowd the city does not really have a lower income neighborhood. The only people that you have to watch out for are the drunks, and the Addison police in case you're the one that's drunk. Don't yell at the police officers, and if they give you a command you should do what they say. The police can be strict, but they're still there to protect and serve.
