Assuming you are married to a Thai national and want that recognised by Thailand then as above the local district office 'amphur' is where to go.
If you where married outside of Thailand they may ask for the marriage certificate to be certified by your embassy and translated into Thai. If you intend to get married here then you will need a letter from your embassy stating you a free to marry.
Amphur office will issue a certificate and Kor ror 2 document which can then be used at immigration to obtain a stay based on marriage. You will have to show 400K baht in a bank account in your name to obtain a 12 month stay
Last time I stayed on Koh Larn I asked the owner of the hotel if he was going to file and thus I would need to upon returning. He told me that legally they where not a hotel and they didn't have to file, thus I didn't. Crazy inconsistencies