I presume you store your PDFs in the cloud. If this is not so, skip the next paragraph of my response.
A PDF belongs to an account (email/password), not to a user. If all users use the same account they all can open any PDF on this account. You have given the password to this account to all users, so that they can log in, right? If you want to restrict which PDFs can be opened by whom you need to get separate accounts to separate who can access which PDFs.
Or you can use certificate-based encryption to authorize only a group of people to being able to open a PDF. Or password-encryption with providing password only to people who may open a PDF.
It is a corporate subscription with only one Adobe ID to sign-in with. By default, Adobe is saving to the cloud. I either want to turn of the cloud option or be able to create different Adobe ID's to sign my users in with to avoid users from seeing confidential material.
You can save signed PDFs in any location you want: local drives, network drives, cloud. I do not understand the problem after my previous explanation.
If you are having ALL USERS who logon to the machine use the same Adobe ID, then they will have access to that Adobe ID's PDF files. If you want to shut off all of the Document Cloud feature services, then have a look at the documentation link below and apply the registry entries that are relevant.