差し詰め (さしずめ) is a somewhat formal adverb meaning 'after all / in the final analysis,' framing what follows as the inescapable conclusion of the preceding reasoning. 抱える, literally 'to hold in one's arms,' is the idiomatic verb for 'to have (work, problems, debts) weighing on one.' The frame [noun]しか…ない means 'nothing/nobody but X' — the main predicate is grammatically negative even though the meaning is affirmative in English.