I want to form a sentence which says"I am yours" in a way that I replace "yours" with thy/thine, but I have no idea which one is correct. I've read that each one of these is used depending on what word comes after thy/thine, but in this case "yours" is the last word in the sentence, therefore it isn't followed by anything. I also have the same question for the sentence "I am hers". What word is going to be a proper replacement for "hers", given the fact that it is also the last word?
Thy and thine are archaic forms corresponding to your and yours respectively. Use thy where you would use your (but see note at end of answer) and thine where you would use yours.
Her and hers do not have alternate/archaic forms. Her is used as a possessive the same way my or your is, and hers is used like mine or yours.
(Note: if the noun placed after thy begins with a vowel sound, use thine instead: thy book, but thine eyes. Archaically, the same was done with my and mine, but this is no longer common usage outside of some poetic use.)
Thy is similar to you; it's a determiner. Thine is akin to yours; it's a possessive pronoun. Therefore, the sentence would be, "I am thine."
For this sentence it might actually be easier and sound a lot better if you rearrange it. Instead of "I am thine.', maybe using "To thine is whom I belong."