How to Tell a Captivating Story in Your Patter Without Overdoing It?

Hey everyone! I've been wondering for a while how to tell a story through an effect without overacting or forcing my voice, and still make the patter captivating for the audience?

I came up with an original story, adapting one of Julio's tricks, which was the story of the innkeeper and the four kings, aces, queens, and jokers. The problem is, when I practice the patter, I feel like the storytelling doesn't come across as believable, but I also don't want to deliver it in a monotone. Any ideas on how to fix this?

Thanks!