Background: we filed a state complaint last school year, the district was found out of compliance and required to complete corrective actions by September and submit evidence of having done so to the state. Three months later, they still haven't completed the required actions.

Have other people been in this situation? What should we expect to happen next? Do we need to file an additional state complaint regarding the refusal to complete corrective actions? Or does the state follow up independently if they do not receive evidence that the district followed through? How long does this whole process usually take?

I'd follow up with the state DOE.  If you don't have contact info for someone at the state DOE, you might need to file another complaint.