John Chrysostom, On the Priesthood - Translation of Greek Phrase In Book 3, Section 6, he wrote,


  οἱ μὲν γὰρ εἰς ταύτην, οἱ δὲ εἰς ἐκείνην γεννῶσι. Κἀκεῖνοι μὲν οὐδὲ τὸν σωματικὸν αὐτοῖς δύναιντ' ἂν ἀμύνασθαι θάνατον, οὐ νόσον ἐπενεχθεῖσαν ἀποκρούσασθαι· οὗτοι δὲ καὶ κάμνουσαν καὶ ἀπόλλυσθαι μέλλουσαν τὴν ψυχὴν πολλάκις ἔσωσαν, τοῖς μὲν πραοτέραν τὴν κόλασιν ἐργασάμενοι, τοὺς δὲ οὐδὲ παρὰ τὴν ἀρχὴν ἀφέντες ἐμπεσεῖν, οὐ τῷ διδάσκειν μόνον καὶ νουθετεῖν, ἀλλὰ καὶ τῷ δι' εὐχῶν βοηθεῖν. Οὐ γὰρ ὅταν ἡμᾶς ἀναγεννῶσι μόνον, ἀλλὰ καὶ τὰ μετὰ ταῦτα συγχωρεῖν ἔχουσιν ἐξουσίαν ἁμαρτήματα. «Ἀσθενεῖ γάρ τις, φησίν, ἐν ὑμῖν; Προσκαλεσάσθω τοὺς πρεσβυτέρους τῆς Ἐκκλησίας, καὶ προσευξάσθωσαν ἐπ' αὐτόν, ἀλείψαντες αὐτὸν ἐλαίῳ ἐν τῷ ὀνόματι τοῦ Κυρίου· καὶ ἡ εὐχὴ τῆς πίστεως σώσει τὸν κάμνοντα καὶ ἐγερεῖ αὐτὸν ὁ Κύριος, κἂν ἁμαρτίας ᾖ πεποιηκώς, ἀφεθήσεται αὐτῷ.»


The phrase in question is «οὗτοι δὲ καὶ κάμνουσαν καὶ ἀπόλλυσθαι μέλλουσαν τὴν ψυχὴν πολλάκις ἔσωσαν». W.R.W. Stephens (link) translates the clause as,


  but these others have often saved a sick soul, or one which was on the point of perishing


According to the Greek text, is the soul that is saved "sick or on the point of perishing"? It seems to me that the soul is "both sick and about to perish." Which translation is more in accordance with the Greek text?



References

John Chrysostom. On the Priesthood. Trans. Stephens, W. R. W. Nicene and Post-Nicene Fathers, First Series. Vol. 9. Ed. Schaff, Philip. Buffalo: Christian Literature, 1889.

John Chrysostom. Regarding the Accusation because He Hid from the Priesthood (Πρὸς τὸν Ἐγκαλοῦντα ἐπὶ τῷ Διαφυγεῖν τὴν Ἱερωσύνην). Patrologiæ Græcæ Migne. Book 3, Ch. 6. Ed. Migne, Jacques Paul. Vol. 48. Petit-Montrouge: Imprimerie Catholique, 1862. (646)
και is simply a conjunction and the context dictates how it can best be translated into English to best convey its sense. While it can simply be translated as 'and' in both places, the OP is correct in noting that typically καὶ...καὶ implies some sort of emphasis (and...or, both...and, on one hand...on the other, as well...as, not only...but also, etc.). The specific emphasis must be determined from the context. BDAG (f) defines:

καί … καί both … and, not only … , but also (Synes., Dreams 10 p. 141b καὶ ἀπιστεῖν ἔξεστι καὶ πιστεύειν.—B-D-F §444, 3; Rob. 1182; Mlt-Turner 335) connecting single expressions Mt 10:28; Mk 4:41; Ro 11:33; Phil 2:13; 4:12. κ. ἐν ὀλίγῳ κ. ἐν μεγάλῳ Ac 26:29. κ. ἅπαξ κ. δίς (s. ἅπαξ 1) Phil 4:16; 1 Th 2:18. Connecting whole clauses or sentences: Mk 9:13; J 7:28; 9:37; 12:28; 1 Cor 1:22. Introducing contrasts: although … yet (Anthol. VII, 676 Δοῦλος Ἐπίκτητος γενόμην καὶ σῶμʼ ἀνάπηρος καὶ πενίην ῏Ιρος καὶ φίλος ἀθανάτοις ‘I was Epictetus, a slave; crippled in body and an Iros [a beggar in Hom., Od.] in poverty, but dear to the Immortals’) J 15:24; Ac 23:3. καὶ … κ. οὐ Lk 5:36; J 6:36. καὶ οὐ … καί 17:25; κ. … κ. now … now Mk 9:22. On τὲ … καί s. τέ 2c. Somet. w. ἤ q.v. 1aβ.—HCadbury, Superfluous καί in the Lord’s Prayer (i.e. Mt 6:12) and Elsewhere: Munera Studiosa (=WHatch Festschr.) ’46.

Smyth's Greek Grammar (2877) indicates that καὶ...καὶ places emphasis on each element separately, which seems to fits with the OP's proposed translation of &quot;both...and&quot;, however Stephens' translation also conveys the same idea effectively in my opinion.
Two alternate translations render it like so:

&quot;...but these frequently save the soul when it is afflicted and ready to perish...&quot; (Cowper, 1866, p. 68, read on archive.org).
&quot;...but these have often saved the sick soul that is about to die...&quot; (Moxon, 1907, p. 67 read on archive.org).

The contrast in the context of the passage is between natural parents vs. spiritual priests/fathers and their ability to

prevent their children from dying
repel disease

In contrast from natural parents, spiritual priests/fathers often saved the soul that

is about to die
is sick

While the order is reversed in the clause in question, I believe the direct comparison is clear between these two abilities between natural parents and spiritual priests/fathers, and any translation that makes such a contrast clear is sufficient.

Paraphrasing (with significant interpretive license), this might look like the following:

God has bestowed a greater power upon priests than that of our natural parents. And the two differ as much as the present and the future life. Our natural parents bring us into this life, but [priests] into [the life] to come. And [our natural parents] are unable to prevent their offspring from dying, or to repel the assaults of disease; but [priests] have often saved sick [people] as well as [people who were] about to die....