Adobe sign can convert word files (no matter how they are generated) and convert them to pdf files. You can add the signature field to the pdf file from Adobe sign. If you want this to be an automated process that is dynamically create the word file, and then use an Adobe Sign API to automatically convert this to a pdf file with signature; yes it is possible. But your ability to do so depends upon the level of the license you purchase.
It sounds like you might be able to use the tagging capability in Adobe Sign. You can add specific tags to a Word file and these turn into Adobe Sign form fields when the file is uploaded to Adobe Sign (or used as a template). You can find details here: https://helpx.adobe.com/sign/using/text-tag.html