I have not heard the phrase "dampening one's style." I think the person who said it was combining two different idiomatic phrases. This might have been accidental. People's minds play tricks that way. Or he might have been trying to be witty or clever by expressing both ideas in one phrase. As you know, "cramping someone's style" means "interfering with what someone is trying to do." Often, it means make them self-conscious, so they can't be natural. "I'm trying to concentrate on making a golf swing. I make big swings. Please stand farther away. If I'm afraid I'll hit you, it will cramp my style." To "dampen someone's spirit" means "to take someone who is happy and make them sad." "I'd like to rent the movie, 'Pee-Wee's Big Adventure.' It's so funny." "I hate to dampen your spirit, but Paul Reubens, the actor who played Pee-Wee, died just last week." "Oh, what a shame."

It's not a common idiom, but probably means the same