Yes, (私は)あなたのもの is a common phrase, and it is used more than a hundred times in lyrics. You can also use other first- and second-person pronouns, for example 僕は君のもの. Of course this can be one of the heaviest expressions to show your love, but that's up to you.
And it's best to leave it in hiragana, just as the OP suggested. The BCCWJ corpus returns 57 results for あなたのもの and no result for あなたの者.
物【もの】 usually refers to inanimate things. 私はあなたの物 is not entirely wrong, but it may look similar to "I'm your gadget" or something. It's possible to say 私はあなたの物じゃないのよ in an angry tone, though.
者【もの】 at least refers to a person, but 私はあなたの者 looks even weirder to me in this situation. A Google search for あなたの者 primarily returns several results from Christian Bibles (which are full of uncommon and archaic expressions), and they're basically used in the sense of "your follower/believer".