Osmanli was the name they used, and that populations under them used too, which btw, just means "of Osman" or "Osmanish" if we are trying for a modern english translation. The problem is that that will open a can of worms: if you're calling the turkish state Osmanli, then why not calling Hungary, Magyarország? Or Austria, Österreich for that matter? The fact is that Osman, or Ottoman if you like the arabic version better, is a dynastic name: it's not possible to avoid this fact. Only with Ataturk's Republic that state became a nation: until then it was just "all the lands under the Osman dynasty rule".
So, leave it as it is, or if you REALLY have to change it, put a dynastic tag to it. Either way, just it's not possible to make all happy: even in medieval times, people called that state as they liked best or as it was more convenient for them.