Are you signing in a browser or stand-alone Acrobat? Which Browser/version, which Acrobat version? What do you mean by "add a digital signature to a document right after creating it"? Do you mean that you created a signature form field and then you try to sign PDF using this signature field? Can you sign using "Sign Document" and dragging the rectangle for the signature?
Please, provide verbose explanation of your workflow, explaining as many details as you can.
I an using Acrobat Pro 9 and it is stand alone. After I create a document out of Word and it becomes an Adobe PDF I want to use the signature button to drag the blue box over a line I have for a signature and place a digital signature that I have stored in Acrobat on the line. When I do that, it says it is read only and it will not save it that way. If I close the document, attach it to an email, and then open it and place the digital signature on it then, it will be attached to the email as it was signed, however it does not save on the computer as it was signed. Adobe used to let me sign and save it right away like I am trying to do.
When you sign with a digital signature the workflow asks you to save the file before the signature process completes. Have you tried to enter a new filename at this step? It is generally a good practice not to save the signed PDF as a different file and not in place of the original one as you may need the original PDF later. If your problem persists when you enter a new filename, then it might be specific to your machine, like the folder where you try to save the signed PDF (or some parent folder) is read-only.