Copyrighting my photographs: Apply a symbol to the shots or file with US Copyright Office?

Any creative act is copyrighted by default. Now and then a creative height is required, a family snapshot of the dog might not fit in. But as a rule, it is very little that is required for copyright protection.

Registering with the copyright office is just a matter of proving that you are the creator.

Adobe Stock DOs & DON'Ts.

https://helpx.adobe.com/stock/contributor/help/editing-dos-and-dont.html

You never see watermarks on images printed on magazines, calendars or t-shirts do you?  That's because watermarks are unacceptable for commercial use.

Stock customers come here looking for professional quality images they can license for use in commercial projects like TV/film, marketing materials, posters, menus, merchandise, etc...  When you submit to Stock, you're giving permission to use the photos but you're not selling your copyright.  And you may submit the same image to other Stock agencies if you wish. 

You MUST NOT add text to files you send to Stock. It will be automatically rejected. 

Copyright is automatic, by the act of creaton. Most photographers neither add metadata nor do they register, and Adobe certainly do anything.  Your shots ARE copyright, just because you took them. Of course keep the originals so you can prove this in a dispute.