I am not sure that the dragon occupies much in way of role in modern American culture. The dragon is more of a connection, or link, to that of older times. Settings of antiquity in which mythological condition so of the dragon were evident are not entirely applicable to the United States. If the dragon does occupy a role in modern American culture, it would be to reflect on its role that it occupied in times that have passed. It is used as a marker of cultures older and more ancient than that of contemporary settings.
The snake is seen in a similar light. I think the symbolic or mythological connections to the snake are not as evident in contemporary American society. I do think that a modern element of the snake that is seen in small sections of the contemporary setting would be in the area of religion. Snake handling, a form of worship in rural and Pentecostal form of worship, is a practice where a preacher or leader of congregation handles venomous snakes in order to prove their protection from God. The snake is seen as a form of evil, and in handling it, there is a clear connection that evil can be handled with the protection from the divine.