Did the Perushim Believe in Gilgul?

Shadal (Shmuel David Luzzatto) says no, in his vikuach al chochmat hakabbalah (my Shavuot reading). He says that this is an adoption of a silly idea by the later kabbalists from others such as the Gnostics.

Yet in a recent comment on this thread, someone points out a curious statement of Josephus which implies the Pharisees believed in gilgul, and cites the Zohar to back this up.

From Josephus:

They say that all souls are incorruptible, but that the souls of good men only are removed into other bodies, - but that the souls of bad men are subject to eternal punishment.

This thread at the department of Greek and Latin at Ohio State University tries to investigate this.

I am no expert on Greek, but I do not believe that gilgul is intended.

Rather, what is meant is merely techiyat hameitim, resurrection of the dead. A person passes away and the body turns to dust, and the soul is removed away, and eventually placed in a new physical body. Not to animate a different individual, but the same individual. Thus, they are eventually moved into other physical bodies. See that the contrast is to souls of bad men who are punished in Gehinnom.

Even if it meant gilgul, the gilgul we are familiar with encompasses that of "bad men" as well, so it would not be the same thing. But it does not mean gilgul.

Further proof of this is that Josephus in this paragraph intends to contrast the Pharisees with the Sadducees. We can see what he writes of the Sadducees in contrast.

14. But then as to the two other orders at first mentioned, the
Pharisees are those who are esteemed most skillful in the exact
explication of their laws, and introduce the first sect. These
ascribe all to fate [or providence], and to God, and yet allow,
that to act what is right, or the contrary, is principally in the
power of men, although fate does co-operate in every action. They
say that all souls are incorruptible, but that the souls of good
men only are removed into other bodies, - but that the souls of
bad men are subject to eternal punishment. But the Sadducees are
those that compose the second order, and take away fate entirely,
and suppose that God is not concerned in our doing or not doing
what is evil; and they say, that to act what is good, or what is
evil, is at men's own choice, and that the one or the other
belongs so to every one, that they may act as they please. They
also take away the belief of the immortal duration of the soul,
and the punishments and rewards in Hades. Moreover, the Pharisees
are friendly to one another, and are for the exercise of concord,
and regard for the public; but the behavior of the Sadducees one
towards another is in some degree wild, and their conversation
with those that are of their own party is as barbarous as if they
were strangers to them. And this is what I had to say concerning
the philosophic sects among the Jews.

Thus we may deduce that the Pharisees, in contrast, believe in the immortal duration of the soul, and an afterlife with punishments for evildoers and rewards for the righteous. We see Josephus mentioned the eternal punishment for evildoers. We might surmise that he mentioned the rewards in the afterlife when he spoke of them being removed into other bodies - that is, techiyat hameitim.

And indeed, that is more or less the sense one gets from reading various gemaras.