Hello,
I'm not sure if I understand fully your explanations, but here are my findings.
Questionnaire settings: Response options
Type: respond many
Save/Resume answers: No
Create all existing types of questions (radio buttons, check boxes, Essay, etc.)
Scenario
1.- Student answers questionnaire, answering most questions, including an Essay question type then leaves the questionnaire without submitting. By loging out, gets a warning message: "Changes you made may not be saved".
2.- Student logs in, goes to questionnaire: All previous answers have disappeared, except for the text entered in the Essay question.
Can you confirm that that is your problem? That is to say that only Essay questions retain their contents? If so, I think this is not a bug of Questionnaire but a standard behaviour of the text editor used in Moodle.