I believe that there is an idiosyncrasy in the form of these expressions involving seriously. Saying that only verbs that take object complements can take adjectives as separate complements to modify their objects and that This is merely a matter of syntax (form), not of semantics (meaning) is tantamount to saying we use an adverb instead of an adjective in the phrase take someone seriously because that's the way we do it. I have yet to see a definitive list of such linking verbs in English; I know that CoBuild regard blush and run as members, and they stress that English has many productive features (ie parallel constructions often develop over time).
A more logical approach to analysing the construction take oneself too seriously (and related constructions, see below) is to consider the definition of the terms adjective and adverb. The classical (semantically based) definition of an adjective as a word that modifies a noun will suffice, though arguably we should classify noun modifiers, eg car in car park, separately. The strict definition of an adverb as a word that modifies a verb (string) is the one I'll select, leaving aside degree modifiers (eg very), sentence modifiers, prepositional phrase modifiers....
Consider The teacher marked the question wrong. Wrong is a predicative adjective, modifying the question (though actually referring to the teacher's written assess ment of the answer).
Contrast The teacher marked the question wrongly. Here, wrongly is an adverb, describing the (accuracy of) the teacher's marking / assess ing.
In the striker shot wide, it is arguable whether the shooting process or the end result is being modified, so it is arguable whether wide is a predicative adjective describing the resulting state, or an adverb describing the shooting process.
In We considered you seriously (for the post), seriously is obviously an adverb (of degree or arguably manner). The considering is what was being carried out seriously (allegedly!)
In We considered you frivolous / too frivolous / too serious (for the post), there are obviously predicative adjectives referring to you.
In We took your application seriously, the adverb seriously refers to how we ascribed importance to (ie took) your application - ie how responsibly WE were operating in our appointments procedures.
But in I can't take him seriously, there isn't a comment on my assessing as such, there's a comment on him: I think he's stupid. We could contrast I considered him, stupidly (though this is actually a comment on the stupidity of my even considering him rather than how stupidly I was thinking at the time I was considering him! - stupidly here is a sentence modifier rather than a true adverb) with I considered him stupid. The people who've felt uneasy with the use of the adverbial form, seriously, in constructions where the allusion is to the object rather than the process described by the verb are quite correct - this is a perverse usage, now fossilised and accepted as correct.
Secondly, there are quite a few verb + adjective idiomatic expressions, probably best treated as idiosyncratic single units, like verb + adverbial particle and verb + prepositional particle constructions, etc, eg
V + AP: take off (of a plane); come up (= occur)
V + PP: take off (a coat etc, or = impersonate); look up (in a book)
V + AP + PP: look up to; pick up on
V + Adj ('intransitive'): take ill; stand easy; come clean; hold true; keep mum; box clever; fall asleep
V + Adj ('transitive'): let loose; force open; lay waste
V + Adj + P: get clear of; make free with; make certain of/that
There are even some quirky verb + noun constructions that stand apart from conventional verb + direct object; eg for catch fire, contrast:
The police caught Tom. Did they really catch him? I know Tom was nearly caught by the police last year.
with:
The shed caught fire. *Did it really catch it? *I know fire was nearly caught by the shed last year.
And even one or two verb-verb constructions, eg make do, let go, let slip, hear tell
According to Claudia Claridge in Multi-word verbs in early modern English, these constructions are all better simply considered as multi-word lexemes rather than subjected to confusing analysis.
Best Answer
It depends on context, but 'take care' is a standard goodbye. Be careful would be appropriate if Jack was going to do something dangerous, and Sarah wanted him to stay safe. Take care technically means the same thing, but idiomatically is used as a way of saying goodbye.