No need to panic. You're not reaching out to The New York Times. That is definitely A goal but it's not THE goal of this script.
Press is the best way to build your authority status as a creative freealncer. It’s a win/win for everyone. The newspaper has cool content and you have something to brag about on social media and on your website.
And yes, local newspapers and magazines are dying for any good content they can get their hands on. You can't imagine how sick and tired they are of having to do yet another piece on Uncle Arnie's Annual Halloween Haunted House and the high school's 27th production of "Our Town".
By you reaching out to them you make their lives soooooo much easier.