The Tur says Ashkenazim recite the passage from Pinchas after the Tamid of the korban mussaf of Rosh Chodesh, but sepharadim don't. The commentators say sephardim don't bother because they will hear it later in leining anyway.

If that's true, then what about those who recite the korban musaf of shabbat on shabbat? If they don't say the korban musaf of RCh [presumably because they will hear it during leining] but typically say the korban shabbat [which is normally not leined], if it's shabbat rosh chodesh [in which case the musaf for RCh will be leined, but also the mussaf for shabbat], wouldn't it stand to reason that they would then omit saying the korban shabbat that day?

Also, for those who recite the korban RCh on RCh, is it appropriate to say it on Rosh Hashanah, or do we purposefully omit mention of RCh on RH?