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What version of JB did you install? If you installed a version that was not intended for your country then it will be a mess.
So it isn't the stock ROM. Have you cleaned cache and dalvik? Do a factory reset as well. Also, why don't you try the stock JB ROM released in Poland?i've installed this firmware Android-4.1-JellyBean-GNex-OTA-Port
i've installed this firmware Android-4.1-JellyBean-GNex-OTA-Port
The fact is you are using an ancient port from another device. You flashed an old port for the Galaxy Nexus with different drivers. Of course its not going to work.
There are plenty of Samsung Jelly Bean ROMs and AOSP (G nex - like) for the S3 that everything works on. I think perhaps you didn't do any research before you jumped in with both feet.
