If I understand you correctly, you start with without_signature_fields.pdf, use Adobe Acrobat to add signature fields, and get with_signature_fields.pdf as result.
In that case Adobe Acrobat as side effect does some changes in the pre-fabricated appearances of the existing text fields, probably meant for optimization, not changing the visual outcome. These changes aren't wrong per se, and other PDF viewers I have quickly available (Foxit, Chrome) still correctly display the text contents.
Apparently, though, Adobe Acrobat outsmarted itself with these changes, probably because those changes make the pre-fabricated appearances not match the default appearance template string anymore, probably because of some other reason. Whatever the reason, Adobe Acrobat does not use the (changed) appearance from within the document to display the text fields but instead something else (probably mangling the default appearance template and the not matching appearance somehow) which fails.
One possible work-around: You say you already use iText to fill the PDF. You could use iText to also create the signature fields at the same time. There is no reason to assume that iText would have the same side effect on text field appearances.
(And iText can also do the encryption...)