You're going between 3 (!) distinct expressions that look similar but are functioning differently.
First, in the relevant Merriam-Webster entry, here is the example sentence:
he had just gotten married when he was shipped right off to war
Right off is an adverb that describes when or how he was shipped to war: right away or immediately.
However, neither of the two sentences found "on the web" under the Merriam-Webster definition fit this usage. The text under these sentences explains that these are selected automatically and not by editors:
These example sentences are selected automatically from various online news sources to reflect current usage of the word 'right off.' Views expressed in the examples do not represent the opinion of Merriam-Webster or its editors.
They allow you to send feedback with a link under the disclaimer. I suggest you do so. In brief, here's what's going on with those "on the web" sentences.
So, second, in this example,
From Burgin, where lanes of traffic in each direction are separated by a median, motorists will be able to make a right turn onto the bridge, and a right off the bridge.
the presence of the article a leads one to read a right as a noun (Merriam-Webster "right," noun, def. 5e) indicating a right turn and off the bridge as a prepositional phrase indicating further information about the direction.
Third, in this sentence,
To a Muggle, there's nothing better than the smell of a fresh Harry
Potter book right off the shelf.
right (Merriam Webster "right," adv., passim) is an adverb meaning something like "directly" or "immediately," and "off the shelf" denotes where the book came from. "Right off the shelf" is also a common expression (see Google Books) that connotes immediacy and easy access from a retail shelf.