My experience has been the docs have a preference for a course of antibiotics first, THEN do the draining/cutting. Except for emergency situations where, of course, they’ll go in first. But for me it was abscesses, which are infections, rather than cysts, which may not involve infection. Basically, I’m not a doctor, I’m not you, I’m not giving medical advice just telling my own story, and if you have a problem seek an actual medical professional.
Here is a link to the beginning (because it wound up being multi-part) to Broomstick’s Most Epic Abscess Story. It’s not the only one, just the biggest.