You get the Marsh badge from defeating Sabrina, who is the gym leader in Saffron City. Sabrina uses psychic Pokemon.
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To find HM 05 Defog, go to the Great Marsh and pick up the pokeballs on the floor until you get it. You'll need the Relic Badge and the Fen Badge to use it. (AKA, Gym badge 4 and 5)
you go to fushia city to battle koga and claim your marsh badge
GET THE LAST BADGE......
go to pallet town and swim down to cinabar island and get the key in the mansion and go the gym and unlock the door and there you go
To get the last gym badge you have to go to cinnabar island and you will meet blue. he will tell you that he will be waiting in veridian city to battle you. when you go to veridian city you go to the gym and fight him at the end of the gym.