Welcome to FT!
Early July is a very busy travel period so I wouldn't be surprised if the flights are indeed full. Usually a small number of people fail to show up for their flights. EK takes this into account when opening seats for sale but if EK miscalculates it is possible that a flight that was 'fully booked' could go out with empty seats.
On 11th July I can't see a single economy seat for sale (via DXB and JNB) I do see seats on the 10th night though, leaving at 2215 from MXP if that would work.