Any support that is acting like a 'joist' is a stringer. They don't need to run bow to stern. Yo have a rotted stringer.
1) Remove ALL rot
2) Identify the source of the leak
3) Fix leak
4) Fix stringers, sole and any rotted bulkheads.
That stringer would normally be encapsulated in fiberglass and resin. It doesn't appear to be. It looks like wet PT.
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