Let's get the transcription right first:
(As so often with the form 'not only... but also', the writer makes the mistake of not observing parallelism:The insurance payout not only would've eliminated Craig's debt
sbut also allowed him to continue his extravagant relationship with Summer and still not have to work for a living.
[He did have debts, but 'debt' here means 'indebtedness'.]
the payout would have not only [note the position] eliminated... but also allowed...
He could have said "The insurance payout not only would've eliminated Craig's debt..." but then he should have continued "but also would've allowed...").
The correlatively conjoined verbs (as far as I understand the terminology - I guess it's possible that someone who'd swallowed a thesaurus might call 'not only... but also' a 'correlative conjunction', but I'm not a connoisseur of this sort of hocus pocus ) are 'would've eliminated' and '[would've] allowed'. 'To continue...' and 'not have to work...' are the complement of 'allowed him'. I don't think these last two are anything to do with the correlative conjunction. But you might like to ask this question in the Diagramming... forum, which is frequented by people with more technical expertise than I have.