They did genuinely believe in the gods. For example, before battles the Romans would use priests to look at entrails to see if the battle would go their way, or if many would die. An eagle soaring over the field was a good omen. Later during the Empire, the emperor was literally worshiped as an incarnation of Jupiter.

Many household had their own "patron" deities, many of which were shared between regions and and cultures. These were not powerful in the sense of someone like Jupiter or Mars, but would bestow blessings or give luck for certain offerings, whilst the main pantheon would hugely impact the world for believers, such as carrying a battle to a certain conclusion or helping someone rise to power.

I was just yesterday rereading Paul Veyne's "Did the Greeks Believe in Their Myths?: An Essay on the Constitutive Imagination", which I love.

It can be fairly philosophical - part of the book's purpose is exploring why different conceptions of Truth make this a complicated question. It certainly has plenty of analysis of ancient texts to show the contradictory ways in which people interact with their ideas of myth, truth, history etc.

It might not answer the question you are asking, but it should give you plenty of others to ask.