Sounds crazy? What about this: Sigismund Vasa is pragmatic enough to convert to Lutheranism (but, like father before, try to make it very 'High Church'). Johan Vasa before his death is afraid about future of his line on the throne and decides to eliminate his brother Karl (none of Gustav's sons was 100% sane, they tended to be cruel and suspicious, and Johan already eliminated one brother before).
Meanwhile in Poland: Jan Zygmunt marry Maria Maximiliana of Bavaria, have 3 kids with her:
-Zygmunt August (1576-1581)
-Jan Stefan (born 1578)
-Kazimierz (born 1580)
Maria Maximiliana dies giving birth to Kazimierz, Jan Zygmunt remarries to his cousin Anna Vasa, daughter of Johan in 1584 (at this time PLC and Sweden have good relations due to common Muscovite danger), they have one son, Zygmunt, born in 1585. Jan II Zygmunt dies in 1597. Is succeeded in PLC by Jan Stefan and in Transylvania by Kazimierz/Kázmér.
In Sweden Sigismund have no surviving children, and his half-brother Johan dies early (Sigismund in absolutely good will brought some excellent doctor from Western Europe, who decimated Vasa family with his 'wonderful' therapies). Say in 1605 Sigismund Vasa's health deterioriate, King calls for his sister (who even as wife of Polish King remained Protestant) and nephew, who are his closest relatives and only remaining legitimate descendants of Johan III. After death of Sigismund I Vasa, his nephew, Sigismund II Zapolya takes Swedish throne, 3 Zapolya brothers rule Sweden, PLC and Transylvania.