I've noticed that behavior and hated it so much that I once started ranting about it to a coworker. He pointed out that when IE (or any other Microsoft app) decides that it really wants to highlight the entire word instead of just starting at the selection point, you can simply drag back over the auto-selected text so that the auto-selected text is no longer highlighted, then go back to dragging over the stuff you want to cut and paste, at which point it should behave like you expect it to.
But I feel sorry for you not being able to use Mozilla.