Yes, they did.
from English SE:
Campbell's book, which is subtitled "Teaching in a Maximum Security
Prison," seems to be mainly about the experience of being a teacher in
that setting, rather than particularly involving work with condemned
prisoners. Nevertheless, the usage cited in Robin Hamilton's answer is
exactly on point. Here is a fuller version of the excerpt that appears
in that answer [combined snippets]:
A student came up with a theme entitled: "Dead Man Walking." It
described a scene he had witnessed in the yard at San Quentin. When a
man on Death Row had to leave the compound containing the gas chamber
for a court appearance, he had to walk across the yard, surrounded by
six guards. The condemned man was dressed in brown, in contrast to the
blue denim of conventional inmates. The condemned man walked with his
head bowed as the loudspeaker boomed out repeatedly, "Clear the yard.
Dead man walking. Dead man walking."
I encouraged students to develop their own writing projects, and if
they got something going would excuse them from regular assignments.
From this excerpt, it appears that the slang phrase was already in use
in the prison at San Quentin (California) by 1978. And if Campbell was
writing about his teaching experiences at a different prison, the San
Quentin incident may have been considerably earlier than 1978.
Actually, back in the day, when I first started out they used to do
that in San Quentin. That is of course where the State of California’s
death row was located. But they used to do it anywhere those guys
went. If they went to a visit or the clinic or to see their attorney
or cross the yard for any number of reasons. They would come out with
their feet cuffed together (a small ass chain in between so they had
to sort of “shuffle” instead of walk), their hands would be cuffed to
a chain which was wrapped tightly around the waist. Each hand in
almost a prone position at the guys side. They had another chain which
would be wrapped around the middle in the back and that would be used
like a dog leash. There would be 4 cops escorting one guy to some
appointment and whenever you heard, “DEAD MAN WALKING,” you were
required to get out of the way. Don’t know if they still do it that
way. Haven’t been to San Quentin in ages. Last time I was there (in
2005 I was sent there to attend court at the Federal Courthouse in San
Francisco), I didn’t hear them say that once. So maybe because of the
politically correct times they have been told not to use that kind of
insensitive terminology. However, they probably still say smart shit
like “back up y’all…condemned dude comin through!”