I don't see how the Ajax mode in the survey could impact the behaviour of the GUI in the backend, so this should not be an issue.
So to reorder question groups I usually would drag and drop them on the left side "Survey Structure".
However, certain conditions might prevent you from reordering groups, so you might have to check if there are any conditions that prevent you from reordering.
E.g. you have the groups in the following order:
1. Group A
2. Group B
3. Group C
And group C has a condition based on questions in group B. Then you can't move Group C before group B, because a condition can only work with questions that are located before the group, not after.
Here what the manual says about that (from your link):
Attention : Questions are displayed according to the order you set from the "reordering panel". However, conditions might prevent the reordering of the questions. In this case, you will have to delete or change the respective conditions first.
So this might be a potential issue. I would also not complete want to exclude the possibility of a bug in your version, which is already relatively old, I think we are currently at 3.19.x. What you could do is to export your survey and upload it at the demo here and see if you also can't reorder the groups.
But first check the conditions.