No, that rumor has been thoroughly discredited. Check Snopes for details.
The switch you're thinking of is cane sugar for high fructose corn syrup, a change that was already well underway. Before New Coke's introduction, a number of bottlers had already converted, with some few still using the older cane sugar sweetener. After the resumption of original formula coke, all bottlers had completed the transition to corn syrup, but it was already happening with or without New Coke.
The formula for Coke has nothing to do with the sweetener. New Coke or Classic Coke had nothing to do with whether or not corn syrup was used. The Coca Cola company produces ONLY unsweetened concentrate that it ships to franchised bottling companies, which then add carbonated water and sweetener to make the final product. New Coke was a change to the formula of that unsweetened concentrate itself, and Classic Coke was a return to the original forumla.