That's actually not quite the same ø. The Swedish one is slightly lighter. Use the deeper Danish ø-sound in a Swedish context, and you clearly sound like a Dane speaking Swedish.

The same is true for ä and æ. The latter has a broader sound while the former sounds a bit closer to e. It is easy to spot Norwegians talking Swedish (such as myself :)

You try to talk Swedish rather than just speaking Norwegian? My experience is that Scandinavians just speak their native language and expect that other Scandinavians should understand them. Any attempts to speak the other language end up sounding like parodies rather than serious attempts.

Not sure it's different enough to warrant a new letter, but Spanish O and Scandinavian Å has the same sound so I guess it's all very muddled.