Andy, I will offer another idea for you.
In moodle, click on the quiz and then in the Administration block, click on Responses. Then download the responses to an Excel file. This file will give you quite a bit of detail about the students and their responses. I am not sure if you have already done this.
Once in this form, you can use Excel or even Word to explore your data in deeper detail. But this method will give you a chance to look at the data, and then decide how much deeper you want to go. If you are using randomly selected essay questions, it will take more effort. I do not know of any easy way to have moodle show you all the responses to one randomly selected question, but maybe someone else here will suggest a method. It might be in the quiz "statistics."
Yes, it can take more effort to match up the wording of the question with these responses, etc., but you can focus on a question, sort by this column, use Excel to break a column into its words, and explore for patterns. I have never done this, but you might be able to take this data into an anti plagiarism tool, or data-mining tool, to help you find patterns. This data give you the students' names (of course), and date information. So you might be able to match some date specific patterns.
When I give online exams, which are supposed to be taken individually, I use moodle to look at the "time" statistics, results, and the students' IP addresses (which are easy to get to in moodle's logs.) If is see two students taking the exam from the same IP address, at about the same time, and with similar results, I call them on "cheating." This is about the best that I can do, and yes, I have caught students cheating.
You have no way to prevent students from printing their exam questions and passing them on to other students. If you are ever showing them the answers, these too can be printed and passed on to others. I am surprised that all of the answers to these questions are not already somewhere on the Internet.
On my exams, I use randomly selected (yet similar) questions, I used calculated multi choice questions, and I use (short) time to try to assess what the students have learned. My current challenge is to create more questions in my question bank. I do not use essay questions.
If you do not end up with an acceptable solution, make sure to go to the Moodle Tracker system and post a feature request. Then, make sure to encourage folks to vote for your request. If you have a good idea that others need, it will eventually be implemented. For example, I have one feature request that I made over two years ago, and it looks like it will soon be provided. Moodle's quiz engine developers are really good!