18:10a As surely as the LORD your God lives,

As it is true/certain that Yahweh your (sing) God lives,
-or-
May the living God Yahweh, the God whom you (sing) serve, be my witness that I speak the truth.

18:10b there is no nation or kingdom where my lord has not sent someone to search for you.

no nation or kingdom has been omitted/neglected by my master in his search for you. (sing)
-or-
My master ⌊the king⌋ has sent his men/servants to every country ⌊in the world⌋ to try to find you. (sing)

18:10c When they said, ‘He is not here,’

Each time they said, ‘No, he/Elijah is not here,’
-or-

⌊The rulers of⌋ those countries ⌊always⌋ denied that you (sing) were there.

18:10d he made that kingdom or nation swear that they had not found you.

my master demanded that they swear that you (sing) had not been found there.
-or-
But my master ⌊always⌋ required/ordered them to swear an oath that they had not found you (sing) .

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