"I am a father."
Jestem tatą. "Tatom" is plural dative (dzięki tatom = thanks to fathers? this form's not very likely to be used...)
I really don't want to sound rude, but be careful for such mistakes: changing ą to om is among the ultimate orthographic mistakes, if someone didn't know you're a foreigner, that would make you look like someone who barely finished primary school.
having trouble distinguishing between the use of tata and ojcem... I am a father, I am a dad...can't ever seem to choose the right one, or see the difference or when to use which...Of course, father is more correct and Dad is more colloquial/personal, but ...there must be a trick somewhere...and I suppose the same would apply to Mum and mother. Sigh....