One part says, that if you are a family member of EU Citizen, you do not need any work permits or oswiadczenie na prace cudzoziemca.
Correct, you don't need it. You can work freely without it, but you should have the Karta Pobytu in your hand to prove your right to work. It won't take long to come through as a family member of an EU citizen, perhaps 3-4 months at the most.