Looking at the evidence base for CBT for health anxiety, it is considered to be effective overall. It's definitely a good option if you are considering treatment.
Is it just health anxiety or other more general anxiety you suffer from?
I also have experienced health anxiety, but also a more persistent anxiety that seems to come from no where, and have found CBT helpful, but also have tried other therapies to support my other anxieties that are rooted in childhood/trauma factors.
Medication tends to help in the immediate, but not address the core issues, so completely understand your reluctance.
Accessing psychological therapies can be hard, as it is assessed on severity, but you can self refer, and there are online courses which are also effective.
How much is it disrupting your life? Depending on this will determine how quickly you can access treatment. It is always worth getting the ball rolling.
But it's a massive step to make the decision to get help, or even consider it, so that's a real achievement!