Not necessarily. If the future monarch decides to use the same crown, then no need to change the badges. This business of "King's" and "Queen's" crown is a bit of a misnomer, I understand. The monarch is crowned with the Imperial State Crown.
The crown depicted on badges and buttons up to the end of Queen Victoria's reign is known as St Edward's Crown. Victoria disliked wearing this crown as it was too heavy and a new crown, known as the Imperial Crown was made for her when she was proclaimed Empress of India. Confusingly this is the one popularly known as the "king's crown' which appeared on badges, buttons etc from the beginning of the reign of King Edward VII until the end of King George VI's reign. The queen's crown now shown is in fact a modified version of the St Edward's Crown as used on insignia in Victorian times. My guess is that the future King Charles (or whatever) will continue to use the St Edward's Crown, but I dont think he will be bound by precedence. Of course lots of army regiments and corps have the royal cipher on their badges and buttons, so they will need to change, as
Roadster points out.