@Travis *******
No. You will *NOT* apply for a Non-O visa. A "Non-O Visa" is almost always a full page sticker that gives you entry to Thailand, generally for 90 days, for "other" reasons. This is almost always issued outside of Thailand, at a Thai consulate, and cost 2K THB (or local currency).
To stay in Thailand after your visa expires, you'll want an extension of stay. For you, that extension of stay is called an "extension of stay based on marriage to a Thai national". You'll get about a
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page stamp with the words "extension of stay". These are only issued inside Thailand and cost 1,900 THB.