I am not trying to be blunt but honestly, even if a GP did "judge" you when you tell them about this addiction, it's a lot better to get help and get off the painkillers than it is to allow the addiction to grow.
Four months at low-level doses is at the lower end of painkiller addiction. The GP probably won't judge you, most are too professional. But the most important thing is getting help now while you still can.
Addiction usually ends in two ways, the user gets treatment and works themselves off the drug, or their addiction spirals until it kills them. The sooner you get treatment the better chance you have of getting clean.
Nobody starts taking opioids or opiates with the intention of getting addicted. The addiction sneaks up on them. Most are in very deep by the time they realise.
A big positive for you is that if you think you are addicted at this early stage, you quite probably aren't. You might be in the habit of taking the drugs and enjoy the feeling, but that's not the same as being chemically addicted. The big sign of an addiction is you increasing the dosage or taking stronger substances. Sticking at the recommended dose, albeit for longer than the 3 days you're meant to, is actually a good sign for you.
(I know it's not the same but I thought I was addicted to soluble paracetamol. I took 8 tablets a day for 20 years and had headaches if I didn't. My doctor told me to stop, and I was able to stop. I had a headache for a week and since then almost never get a headache. I wasn't properly addicted in a chemical sense, it was just a pattern of behaviour that I didn't see as a problem until a doctor told me it was. Even though it was bloody obvious to a normal person it was a problem to be taking painkillers every day for two decades - I needed to be told!)