Defer the draft walk and JSON serialization until the storage write flushes.
Preserve hydration, migrations, attachment verification, and final flushes.

Continues the web portion of [#9049](https://github.com/pingdotgg/t3code/pull/9049).
The mobile storage migration remains separate.

Created with GPT-6 Astra (preview) in Codex.

Co-authored-by: Wout Stiens <71498452+StiensWout@users.noreply.github.com>
Co-authored-by: Claude Fable 5 <noreply@anthropic.com>