Obsequious

Fawning, submissively eager to please and agree

Examples:

Those men are most apt to be obsequious and conciliating abroad, who are under the discipline of shrews at home.
— Washington Irving, Rip van Winkle

She what was honour knew,
And with obsequious majesty approv’d
My pleaded reason. To the nuptial bower
I led her blushing like the morn; all heaven
And happy constellations on that hour
— John Milton, Paradise Lost

Prison taught him the false smile, the rubbed hand of hypocrisy, the fawning, greased obsequious leer.
— Anthony Burgess, A Clockwork Orange

But it is hard to know them from friends, they are so obsequious, and full of protestations; for as a wolf resembles a dog, so doth a flatterer a friend.
— Sir Walter Raleigh, writing about flatterers, in The Voyage of the Destiny.

Etymology:

Latin: Ob = after and sequi = follow. Think “follower”, with sequi as “sequence”