The four soft drink brands mentioned in the crossword solutions are Nehi, DADS, Fanta, and Shasta. Nehi is a brand of soft drinks that was first introduced in 1924 by Chero-Cola/Union Bottle Works. DADS is a brand of root beer that was first produced in 1937 by the Dad's Root Beer Company. Fanta is a brand of fruit-flavored carbonated drinks that was created in 1940 by The Coca-Cola Company. Shasta is a brand of soft drinks that was founded in 1889 by the Shasta Beverage Company.