Both scenarios you suggest are what Gandalf fears if they do not destroy the ring, either:
The ring is kept by someone weaker, like Bilbo or Boromir. They are eventually overpowered by Sauron's forces, and the ring is returned to Sauron who will use it to overpower Middle Earth.
The ring is taken by someone who is powerful enough to wield it, initially for good, but because of the corrupting nature of the ring, will eventually fall and become no better than Sauron.
This second scenario is most clearly shown in The Fellowship, in two occasions when Frodo offers the ring first to Gandalf:
GANDALF
You cannot offer me this ring.
FRODO
I'm giving it to you!
GANDALF
Don't tempt me, Frodo. I dare not take
it, not even to keep it safe.
Understand, Frodo...I would use this Ring
from a desire to do good...but through
me, it would wield a power too great and
terrible to imagine.
And secondly to Galadriel. This scene is particularly telling because it shows how much the ring tempts people with power, and that Galdriel knows that it would essentially possess her, and that she would no longer be herself:
FRODO (V.O.)
If you ask it of me, I will give you the
One Ring.
GALADRIEL
You offer it to me freely...I do not deny
that my heart has greatly desired this.
Galadriel suddenly seems to rise in stature before Frodo's
eyes. Frodo is suddenly afraid of her.
GALADRIEL (CONT'D)
In place of the Dark Lord, you would have
a Queen, not dark, but beautiful and
terrible as the Dawn. Treacherous as the
Sea! Stronger than the foundations of
the earth...all shall love me and
despair!
Galadriel suddenly laughs...a slender Elf-woman once more,
clad in simple white, her voice soft and sad.
GALADRIEL (CONT'D)
(gently) I pass the test.
(laughs) I will diminish and go into the west and
remain Galadriel.
