signatory / signee / signer

The all mean the same: one who signs a document. 'Signatory' is usually reserved for very important documents, e.g., treaties between countries. 'Signee' sounds appropriate for your use. 'Signer' is the least formal of the three.

we see "signatory" a lot in regular legal contracts.

What we usually see in regular documents, letters, etc. is simply:
signed by:

Signee and signer are NOT the same. The suffix -ee refers to the object of a verb, and -er to the subject. In this case, the verb is sign, and so the person who signs the document is the signer. Obviously there cannot be a person who is signed, so signee is impossible.
It is possible to add the -ee suffix to almost any verb, with more or less reasonable results, and in practice it is not now used much in its proper meaning.
Having said that, it is quite common to see the -ee and -er suffixes used in such a way as to make it clear that they are intended to have the same meaning; although this is to be deprecated, it is becoming a common usage. Indeed, use of -ee in its proper way is usually for mildly comic or ironic effect.

I am well aware of the difference between the -er and -ee suffixes. Signee sounds odd to me too. Then again, I don't write the dictionaries.

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sign·ee (sī-nē')
n. One who has signed a document, such as a contract or petition.

The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2009 by Houghton Mifflin Company
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sign⋅ee  
–noun a person who signs a document, register, etc.; signer; signatory: a signee of the Declaration of Independence.

Dictionary.com Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2009ht © 2009 by Houghton Mifflin Company