You may have already tried this.

You can fix the mirror supports pretty easily.  There is a spring that
provides the ball joint tension.  Corrosion freezes the part (ball thingie)
the spring pushes so the tension can no longer respond when the mirror
moves.  Consequently the tension is either excessive or to little.  It can
easily be accessed if you remove the two screw head bolts (the nuts are
welded to the door frame so they won't be lost) and remove the mirror from
the door.  Soak the nut and spring inside the mirror mount with penetrating
oil and when you can get it apart, soak the sliding ball piece with oil
until you can get that apart.  Clean the corrosion up on everything so the
ball will slide easily on the stud and reassemble with enough pre-load
tension on the spring to hold the mirror.  It should be adjustable and stay
in place now.  If it doesn't, remove and tighten the nut to increase the
pre-load and try again.

Of course, if you can't get the ball thing off the stud or if the stud
twists off it is pretty much toast.  This is entirely possible.  You should
be able to find replacement mirrors in someone's junk pile.

Good luck,
Ed