Use a digital signature field instead of Fill & Sign.
I can reproduce my issue even when using the Digital Signature. After I sign and save the file, I just use the print dialogue to "Print as PDF", which turns the digiatal signatures into editable images just like the Fill & Sign method.
You can't avoid this.
After you sign the file you can apply a security policy to prevent printing. Also, if the user does what you described the new file is not going to be digitally signed any longer.
"which turns the digiatal signatures into editable images just like the Fill & Sign method."
No, it doesn't. It really doesn't. This is a common misunderstanding of digital signatures.
A digital signature is not what you see on the page; it is based on a certificate, and can tell with absolute certaintly whether a PDF has been changed since it was signed. If the PDF is changed in any way, the signature is either gone, or invalid. Sure, there is a mark on the page.
Remember, you CANNOT trust marks on a page. With no special software, anyone can "lift" a signature from anywhere and stick it on any document. They can in seconds make a copy of a file and change the signatures. Faking is so easy, that digital signatures are an absolute must. With digital signatures, the fake won't have a valid signature.