Right-click them in Form Edit mode and go to Properties. Make sure they are not set as Locked (under the General tab).
None of them are locked.
Is it possible that whoever created the form fields, did so multiple times, on top of each other?
If that's possible and there is no way to delete multiple instances of the form fields, do I need to keep deleting and saving the document?
Yes, it's possible. It's also possible that there's a script embedded in the file that keeps generating them each time you open it...
Would it be possible for you to share this file with us?
I fixed it. I think there was a script in it, but I tried to delete the information in the create and edit script box but it wouldn't let me.
I've had a small amount of experience with Photo Shop and know a bit about flattening, which I never knew Acrobat also has.
I used that and the form fields don't show up anymore. I'm sure the script is still there, though.
I have no knowledge or experience with Java Script creation.
I can still send you the file if you'd like. Just tell me how.