"I am eating with a plastic fork."
Translation:Jem plastikowym widelcem.
Because nobody says "z" to express using some tool. "Z" means "with" only in sense "together with" (accompanying). To express using some tool/instrument you should use Instrumental without any preposition like in the sentence above.
"sztućcem" (Instrumental of "sztuciec", usually used in plural: "sztućce") is a piece of cutlery. So it's not a synonym of a fork, as a spoon and a knife also can easily be meant by that word.
If you use something as an 'instrument', there is no "z", just Instrumental without any preposition.
Saying "Jem z plastikowym widelcem" would mean that the plastic fork is sitting next to you, eating as well.