When not used as a polite closing, are "Thanks" and "Thank you" sentences ending with a period? "Thanks. I appreciate your effort." "Thank you, Kevin. I will be in touch soon."
closed as not a real question by Mehper C. Palavuzlar, Irene, Barrie England, Hugo, simchona Jan 7 '12 at 23:16
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"Thank you" is a sentence with an implied "I" or "We", as in "(I) thank you."
"Thanks" is a shortened form of "thank you", but you can use it as if it has an implied "You have my" or "I offer my", as in "(You have my) thanks."
Grammatically, 'Thanks' is as much a sentence as 'Yes' or 'Goodbye'; but, as you suspected, it is unusual to put a full stop after it when followed by a related sentence. Personally, I would put a comma in your first example and a semicolon in the second; but only because of the close relation of the meanings, and it would be the same if you expanded Thanks to Thank you.
Personally, I would use an exclamation point (e.g., Thank you, Elizabeth!), but it appears that I'm in the minority. By far, most people seem to use a period. Here's a relevant corpus search (Note: the numbers are representative, not exact. Click on the punctuation marks to see examples of use.)