Ye, sure, the Roman Empire revered Augustus and it also held sacred his children, wife and all his family as well.Particularly in Asia Minor entire cities and communities made oaths, inscriptions, etc, to Augustus, Gaius and Lucius, for their genii and health, and later on to Tiberius etc too.
But there is a great difference between honoring Augustus' family and taking an oath that only his dynasty, his line, his family will rule the Empire. There was never such a thing like an 'official' crown prince of the Roman Empire. So there would never be a situation like "The king is dead, long live the king!"
That's a great difference between the Roman political system and European medieval monarchies for example.
IIRC there was a Roman emperor about a thousand years later after Augustus and that emperor was so worried about his heir and his family that he made all the Empire to swear solemnly that the next emperor will be his son, and that there would be nothing wrong done to his family.
But that was an exception.
Usually the situation was like this:
if the emperor is dead the Romans would naturally prefer someone from his blood line, his family and relatives as his successor. But there were nothing like 'succession laws' in the Roman Empire so that was never a dogma.
If there was a fool on the throne he might break the magic spell of his dynasty and after his death the Empire would like to look for someone not related to the late emperor as his successor. Preferably an outstanding general who had support of the army.