It depends on what you mean by "mushy," be it just wilted leaves, or, as I am envisioning, the kind of mush that turns to sticky foul-smelling goo like a rotten potato when you poke it with your finger.

If it's the kind of mush that easily gives way when you poke it, I would wipe/wash away as much of the mush as possible to the remnants of firm stem (if there is any there) so that the living portion can dry. There may be a very definite boundary between the dead/rotting portion and the living portion.

It could be that it's rotting from the bottom up (they cannot live in standing water) and the only way to save it is to re-root the stems like a cutting. Since the stems are partially succulent, the stems can live quite a while after the roots have died and that may be what is happening. If that's the case, let the stems dry for a day before you root them in soil.

If the top part is rotting but the roots and lower parts are still healthy, you can cut the top half off and the bottom half will regenerate new leaves and flowers.

Tropical geraniums are pretty tough, partly because of their ability to re-generate from any stem (which you know if you've ever tried to propagate/root them from cutting). So yes, you can cut them way back and they live as long as the part you are trying to re-grow is made of healthy, firm tissue and includes growth nodes.