who makes sure that none of the county's money is spent without the county board's approval?

Question

county clerk or auditor or assessor or county treasurer [give source or explain

Answer 1

Which county you referring to??? What spending you referring to?

Each county will have a budget setup during an annual cycle, money is budgeted for a variety of things. Somebody (e.g. treasurer, comptroller, etc.) will track that spend, run-rate etc. and revenue sources too. Approvals for new spend would normally come from county board or based on a voter initiative.

Auditors generally don't do day to day monitoring of spending, they audit for compliance with policy and law etc.

Assessors typically run the property tax collection and assesment arms of the county.

But some of these roles could vary based on how the county was setup, its charter, etc....

You should look at your county's web site or go to the library. I'm sure the by-laws or other governing principles are documented there.