shed skins do not carry the pigments and patterns so it is hard to say. If I had to guess it would probably be a rat snake which are not poisonous. 

Aside from coral snakes, which would not be that large, nor occur in Spartanburg County, all the rest of the venomous snakes would have much fatter bodies in the middle tapering at both ends. non-venomous would be the same width for most of their length., only tapering close to each end.

Is the head area still intact? That would be helpful. Rat snakes are so named because that is one creature they are prone to hunt which means crawling into holes to find prey. If your dryer vent is connected, then I doubt it could possibly be in your house. 

Copperheads are the most common venomous snake in the upstate of SC. They still have a pattern visible on the skin. https://www.reptilefact.com/copperhead-snake.html

3 species of rattlesnakes can be found in the state, but they are usually not found in populated areas. The tail of the skin would be blunt where it ended at the rattles.

I learned something new watching a recording of a recent Making it Grow episode where a naturalist was on the show with two snake skins. Near the tail on the underside behind the "anus" non-venomous snakes will have split ventral scales venomous snakes will have single ventral scales. You can watch this episode S2023 Ep14 | June 27, 2023 found here. https://www.scetv.org/watch/making-it-grow. The segment showing actual snake skins and telling the difference starts at about the 46:20 minute mark in the show which is near the end.