How To Save Fields As A Template For New Documents?

For my clients, each project agreement document obviously is different, though the bottom of each document where the signature fields go is exactly the same. Rather than having to tediously create and configure new fields each time I want to send a contract for signing, is there a way to simply save all fields that can be placed over a new document?

I would think this would be an obvious feature, but of course with Adobe, things have to sometimes be unnecessarily confusing. I am able to save the fields as a "template", but it doesn't 'seem to allow me to change the background document. Am I missing something here? Can anyone chime in? Thanks.