SPRINGFIELD, Ill. — On the grounds of the Old State Capitol here, where nearly 160 years ago, Abraham Lincoln held forth on “a house divided,” Hillary Clinton on Wednesday lamented the Party of Lincoln’s transition to the Party of Trump, casting the present moment as an indelible stain on Republican history.

Yet even as she savaged Donald J. Trump as an existential threat to American democracy, a week before Republicans plan to nominate him for president in Cleveland, Mrs. Clinton set off on a delicate balancing act of her own.

She waded with care into the thickets of national reckonings over police violence and violence against the police, hoping to position herself as an unlikely agent of harmony.

And in an uncharacteristic admission, Mrs. Clinton assumed responsibility for at least a small measure of the fractiousness in the national discourse.