Department of Health (or records, etc) is for Birth Certificates. Secretary of State (or second address in list) is for Apostille.
Call for latest fees, and for the name you need to put on the check (Pay To. Some only accept money orders. CALL AND ASK or check website if listed.
Keep in mind that each state has its own strange laws regarding these. In fact, it actually makes Mexico seem organized in comparison.

I will post this by state because ValueMD will not post anything over 50,000 characters

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"You do not truly understand something unless you can explain it to your grandmother." Albert Einstein