I'm not sure it matters as much whether your audience is familiar with the particular songs you sing as whether the musical genre and style are familiar to them.   Sure, some songs won't make a lot of sense to a new audience unless you give a bit of background. (I realized that recently when I sang Joni Mitchell's "The Gallery" to people unfamiliar with it, without explaining what it was about.) And some audiences expect to be asked to sing along at least part of the time, so familiar songs are well received. But unless you're singing old art songs to a country music audience, Arabic love songs to a Celtic loving group, Balkan music with "non-Western" time signatures to a bluegrass group, etc., I don't think introducing a new song is a turnoff to most audiences. In fact, it can make their ears perk up and get their attention.
If the lyrics are important and non-repetitive, though, your audience may not really "hear" a new song and grasp the story the first time.    But that can generate questions about who wrote it, the title, etc.