It is very much a matter of style, and the remarks by Allen & Greenough are actually misleading - what they mean is not that the adjective functions like an adverb in the Latin (an adjective is an adjective, and it qualities the noun with which it agrees), but rather than WHEN TRANSLATING INTO ENGLISH, you might want to choose an adverb instead. Just as general rule, Latin is more sparing in its use of adverbs than English is, at least in part because adverbs, as indeclinable words, are so "disconnected" from the rest of the sentence.
P.S. Here's the note I included about that in my Aesop's Fables in Latin book (http://tinyurl.com/dbmbg3
Adjectives and Adverbs
In Latin, the system of nouns and adjectives is very strong and flexible, but the adverb system is much less fully developed. As a result, there are often instances where Latin will use an adjective while in English we might tend to use an adverb instead. This is especially true when Latin uses an adjective to modify the subject of the verb. Consider this example from the fable you are about to read: asinus oneri totus succubuit. The adjective totus modifies asinus, the subject of the verb. So, translated literally, the sentence would read: “The whole donkey collapsed under the weight.” That is what the Latin says, but it sounds quite odd in English! If you use an adverb in your English translation, instead of an adjective, the result will sound much more idiomatic: “The donkey collapsed completely under the weight.” So, whenever you see an adjective being used to modify the subject of a verb in Latin, it is worth thinking about whether that adjective really belongs with the noun, or whether it is perhaps better rendered in English with an adverb instead. (Similarly, if you are translating from English into Latin or composing in Latin, think twice before you use an adverb: there are many situations where we might use an adverb in English, while Latin would prefer to use an adjective instead.)
Here is the fable in question:
DE EQUO ET ASELLO ONUSTO. Agitabat Coriarius quidam una Equum et Asinum onustum. Sed in via fatiscens, Asinus rogabat Equum ut sibi succurreret et velit portiunculam oneris tanti tolerare. Recusabat Equus et mox Asinus oneri totus succubuit et halitum clausit supremum. Herus accedens mortuo Asino sarcinam detraxit et, pelle superaddita excoriata, omnia Equo imposuit. Quod cum sensisset Equus, ingemuit, inquiens, “Quam misellus ego, qui, cum portiunculam oneris socii ferre recusaverim, iam totam sarcinam cogar tolerare.”