Cleansers in general leave your skin a little drier, so if having super desiccated skin is of major concern for you, consider ditching regular cleaning altogether and using a washable oil instead (Shu Uemura Gentle Cleansing Oil is the classic).
But to answer your question about soap more specifically, the answer is no—not everything will give you that tight-skin feeling. The people behind Dove Beauty Bar are always quick to talk about how their product is three quarters moisturizer, but there are other soaps designed not to wreck your skin, too. Milk-based soaps like Alpen Secrets Goat Milk Moisturizing Bars are the traditional gentle option, but oil-based soaps like Kiss My Face Olive & Honey are a good alternative as well.