To check a word's part of speech, you need to see which syntactic equivalent you can replace it with, whose part of speech you already know. Picking synoymous expressions is the easiest way: the synonyms of "et" the conjunction are "atque" and "que", the synonyms of the adverbial one are "etiam", "item", "quoque". Conjunctions couple independent structures of the same syntactic level, conjunctive adverbs turn the clause they couple into a dependent adverbial modifier. This adverbial modifier can be replaced with an adverbial clause, such as a dum-clause. Let's take some sentences:
Mārcus et Quīntus discvpulī sunt = Mārcus ac Quīntus discvpulī sunt = Mārcus discvpulus est et Quīntus discvpulus est.
Mārcus, etiam Quīntus, discvpulī sunt = Mārcus discvpulus est, etiam Quīntus = Mārcus discvpulus est dum Quīntus item discvupulus est.
Here it's clear that the rightful place of the etiam-clause would be at the end of the sentence, and placing it right next to Mārcus instead of using the coordinate conjuncton "et" feels clumsy, as if you'd forgotten about Quīntus.
Mārcus dormit et Quīntus vigilat ≠ Mārcus dormit, etiam Quīntus vigilat.
The latter makes Marcus also perform the action that Quintus is performing, equating dormīre with vigilāre, which makes the sentence incomprehensible. In proper grammatical terms I guess the verb in the etiam-clause would be right-dislocated from its rightful place as the top-level predicate with the subject Mārcus.
Mārcus dormit. Quid? Et Quīntus? Hau crēdō! = etiam Quīntus dormit?!
This contrasts with "atque Quīntus?", which, only being a conjunction, serves as a logical transition "but what about Quintus?", or, if we included the verb, "atque Quīntus dormit?" = "whereas Quintus sleeps?", which makes no sense if the verb is the same, which is why it defaults to the "what about" meaning in its absence.
Now for some test examples from the L&S entry for "et":
tē enim jam appellō, et eā vōce, ut mē exaudīre possīs = ...atque eā vōce [id faciō = appellō]
The verb is coordinate. "...etiam eā vōce, ut" would create an adverbial clause "tē appellō, dum item eā vōce appellō", with the meaning "nōn modo hōc modō appellō, sed etiam eā vōce, ut exaudiās", or even "nōn sōlum hāc vōce, sed etiam eā".
advēnit, et nāvibu’ complēvit lītora = advēnit, atque complēvit ≠ advēnit, etiam complēvit
"etiam" turns it from a temporal relation into a dry list of what somone did, and without a conjunction ("atque etiam") feels especially abrupt.
Ubi tunc erās? Rōmae. Vērum quid ad rem? Et aliī multī = etiam/item aliī multī = dum aliī multī quoque Rōmae erant.
Because there's nothing for the noun phrase "aliī multī" to coordinate with in the sentence, "et" cannot be a conjunction. "atque aliī multī" would mark the end of the previous thought and create a new clause that transitions from the topic and which needs completion, something like "and besides".
ego vērō et in ipsā suffocātiōne nōn dēsiī = etiam in suffocātiōne = dum suffocor, tamen..
Again, since "in ipsā suffocātiōne nōn dēsiī" cannot be coordinated with anything, "atque" in place of "et" would start a whole new clause. The fact that "et" can be reverse-rephrased with "tamen" makes it even more obvious that it's an adverb.
cantus nōs nonnumquam et citāta modulātiō instigat = cantus et modulātiō nōs instigat ~ cantus nōs, item citāta modulātiō, instigat, the latter = cantus instigat, dum etiam modulātiō instigat
This one is a matter of interpretation. In the absence of intonation cues indicating that "et citāta modulātiō" is a parenthetical adverbial clause, "et" is assumed to coordinate "cantus" with "modulātiō".
I know I might not be explaining this in the most clear and structured way, but if I had to draw a bottom line, it would be that assuming you get the gist of the sentence, there hardly ever arises a need to consciously determine which part of speech "et" is, as only one interpretation will make sense, and you will arrive at it subconsciously. There might well be cases where it's difficult to determine, but these are also the cases where it matters the least. Still, if the need does arise, I hope the examples above prove useful as a guide.