Unfortunately, as with everything in language learning, there is no way to do something effortlessly! Remember that learning a language isn't just about memorising things; it's about getting used to them -- becoming familiar with a new way of communicating ideas.
If you're having difficulty using the right particles, it might be that you haven't fully understood the differences between them. Perhaps go back to basics: Talk To Me in Korean have helpfully collected together their past episodes on particles which might be a good place to start (http://talktomeinkorean.com/lessons/particles/
Once you've revisited explanations of the particles, try experimenting with them.
As an exercise, you could choose some sentences from something you're reading in Korean. You could use anything with particles in it but it helps if you think the sentence is interesting. Read the sentence a few times, so that you're happy you understand it. Then translate it into Polish.
Once you've done that, write the sentence out in Korean but change the particle. You could replace it with something completely different, but it's more helpful to use a particle that you think is similar. Read your new Korean sentence. How has the meaning changed? If you translate your new Korean sentence into Polish, what is the difference? In your translations, try to write not just what you would say in Polish, but how you can convey all the information contained in the Korean phrase.
Does this exercise reveal parallels between Polish and Korean? Do you find certain Korean particles lead to you using certain expressions or cases again and again in Polish? Is there any part of the meaning that cannot be translated comfortably into Polish and requires two sentences instead of one? Or a footnote/parentheses to capture all of the meaning? Understanding is built on seeing these connections and, once you understand, it will start to feel effortless.