No, they don't. A few do, but the operation is expensive, painful, and not always satisfactory. Many will take hormones and some will have non-genital surgeries, but full gender confirmation surgery is for a minority.

A great many, perhaps a majority, don't do anything to their bodies at all. They change their clothes, their pronouns, their hair, and their lifestyles, but have no surgery or medicines. They live in society as their gender, and anything that's seen only by the people they're intimate with are a subject for discussion between them.

The result is that yes, there are a fair number of men with vaginas and women with penises. Which doesn't matter to you unless you're either their doctor or their lover, but it's important to recognize that it is in fact the case.