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Writing "to" and "from" on gifts in Polish. [41]
Tadziu is the vocative form of Tadzio and perfectly fine in Polish. Tadz must have evolved in your American family. Dimunitives are pretty regular here - especially for such old fashioned names as Tadeusz. If anyone wanted to shorten Tadzio in Polish, it would be Tadź, as 'dz' is softened by 'i' in Tadzio and Tadz would sound too hard. But again there are regular dimunitives for names used in Poland for centuries and I have never come across Tadź in spoken Polish, film or literature. That just sounds like a foreign form of a Polish name. So is Tad - like a mixture of Tadek and Ted. You can't forget your family was influenced by English even if they were born in Poland. It's Ponglish like sklep na cornerze instead of sklep na rogu.
Felikska is a mispelling, I'm afraid. I find it difficult to pronounce and I'm a native speaker. It can't have appeared in that form in Latin Church books. It doesn't sound Polish and a dimunitive of any sort can't have appeared on a birth certificate. It's a formal document. If it's really on her birth certificate, someone must have misspelled it - either a half-literate Polish scribe or an American clerk.
You provided a link to Ms Zaleski's obituary. Her sister's obituary is also online and the name is spelled Feliksa there. And it has just struck me that if you're related to these two ladies, Feliksa's husband was called Stanisław. So perhaps he was called Staś not Tadz, Staszek not Tadzek. Other dimunitives for Stanisław are also Stasiek, Stasiu, Stasio.
Anyway, all the best for you.
Szczęśliwego Nowego Roku!