I don't believe you can disable it, I'm not an expert in this area but have enough experience to know that this is standard in Acrobat/Reader that the existing content is "flattened" (no longer editable) after a signature is applied and the PDF is saved.
The ways around this that I am aware of are by using a workflow tool like Adobe Sign where you can define who fills and signs which parts of the document.
Another method would be for you and the person who needs to add more data after you have signed it to use Adobe Fill & Sign. You could fill and sign the PDF, download and email to the other person who could also open the PDF in Fill & Sign and would be able to add additional text. Fill & Sign does not work with the original real form fields, it basically treats the PDF as an image and allows you to add text, annotations and signatures - it is really designed for a single user but would satisfy the workflow you specified.
https://cloud.acrobat.com/fillsign
https://helpx.adobe.com/document-cloud/help/fill-and-sign.html
Thanks,
Josh