Hope you are doing well and sorry for the trouble. As described you want to add the date stamp to the PDF form when it is completed.
Would you mind elaborating more about adding stamp to ready only field? Are you able to fill the form or is it protected and read only?
I have checked the Adobe account with the email you are using on this community and haven't found any subscription/product registered. Please confirm if you have Adobe Acrobat Reader DC (Freeware application) or Adobe Acrobat DC (paid application) and what is the workflow/steps you are using to add the date stamp to PDF form?
Use a digital signature field. It will show the current date and time when signed. However, you can't rely on this information to be correct, as the user can always change the system date before signing it. If you want a reliable timestamp you need to use a service that issues it, like Adobe Sign.