Can you fix it without breaking my live store?

Yes. Cron and indexer work is mostly configuration and state cleanup, not code that ships to the frontend, so the blast radius is small. We back up the relevant tables before touching them, clear stale rows rather than truncating whole tables, and run a clean reindex that the storefront keeps serving the old index during. If anything looks off, we roll back the state and reassess, your shoppers never see a half-reindexed catalog.