It might be difficult to imagine how those two things are separable if one has lived their whole life with them in congruence. If perhaps, you close your eyes and concentrate on your being, there is a part of you that feels that your sense of manness of womanness is part of who you are? What would you do if you retained that sense, and woke up in the body of the opposite sex and were expected to behave in congruence with that contrary to your internal sense of self? It can be a bit like that.
I wouldn’t want to wake up and be a man, but not for any reasons that are biological. I work in a male dominated field, and most of my interactions are with men, I like the things I can do and get away with, where a man would not have the same experience. The male experience sounds lonely, tough, and a lot of your success seems to just depend on chance and grit. My life has some bad parts, but it’s softer and more comfortable. Would I have dysphoria as a man? I don’t know, to me it sounds like it’d be something closer to envy, but maybe that’s just dysphoria by another name. Maybe that’s the root cause of why so many men lash out at women.
Every trans person's experience is different, I know that the "wrong body" terminology doesn't apply to everyone.
There are two ways to achieve that: changing the body, and changing the mind.
(I'm not claiming one to be easier than the other, but I've noticed most people have a strong preference for changing their body rather than changing their mind)