No, it’s a predicate adjective, and accurate as it stands.
“I eat healthily” translates as “The manner in which I consume my food is one that conduces to good health”, i.e., I chew everything thoroughly, don’t bolt anything, etc.
“I eat healthy” is short for “I eat [such things as are calculated to keep myself] healthy.”
It’s a juncture between subject and adjective that is not drawn by a form of “to be” but reshaped by the verb. “I eat with the explicit goal of good health.”
The “Southwestern adjectiverb” of “Drive friendly” or “Drive safe” is a combo of a regional tendency to use predicate adjective for adverb in colloquial speech and a focus on the unspoken subject: “[You should] Drive [in a manner that shows or pretends that you are] friendly” or “…[in a way that will keep you] safe.”
The form is related to the sensory-verb predication:
“This sentence sounds stupid” probably does, but implies a relationship between stupid-soundingness and the self-referential sentence.
“That dog smells bad” implies he’s been rolling in the mud again, or chased a skunk. Of course, if a nasal condition is messing up his sense of smell, he instead smells badly.
“This fish tastes bad” means you shouldn’t have waited three days to cook it; “This fish tastes badly,” as anyone would if they had a fishhook stuck through their tongue.
The 1950s cartoon character Pogo ran for President in one story arc (to use a 50-year anachronism), with a campaign slogan, modeled on “I Like Ike,” of “I Go Pogo” – the message of support was clear even though the sentence defies a precise grammatical analysis. Whatever “Pogo” is doing in that sentence, it’s certainly not a direct object, but rather related to everything else by one or a multiplicity of implied constructions – the ambiguity in precisely what preposition or phrase one goes for, about in favor of, in support of, etc., Pogo is a part of the construction’s charm.
There’s a genius in American colloquial English that seems to know precisely when to substitute a subject-pointing verb-flavored adjective for the “proper” adverb of formal English, though I think it would take a student of colloquial English some extended study to try to define when it is and is not done.