It's far from universal, but many do it. I sure as heck did.
The thing about clothes is that when women wear men's clothes, it's a mark of liberation, so even if it's transgressive, it doesn't upset people's sense of men's superiority. But when men wear women's clothes, it's a debasement. At least, that's how a lot of people see it. Anyway, that's why I think the asymmetry is there. But there's a limit -- I don't think people see Leslie Feinberg as "cute."
I'm one of those who cross-dressed before transitioning. It's a double-edged sword as far as coping mechanisms go. It was a way for me to privately cling to my feminine identity, but it was also a nasty reminder of what I didn't have. There were other things I didn't do; for example I started to read one of my sisters' copy of Little Women when I was ten or twelve, and I just got disenchanted, because it was describing an experience I was cut off from. I envied my sisters' ballet lessons, but refused to take lessons myself (my mother specifically offered), because it would just highlight my alienation.
We all find ways to try to be true to ourselves and honor our identities before transitioning, and they all come with the potential of making us feel even worse by seeming to be a foretaste of a feast that will never come, and they are all either too transgressive to be acceptable, or so acceptable that nobody notices or cares.