Failing to fully exhale can lead to CO2 build up. The build up is between your lungs and your mouth. If you purge the second stage you are moving air from the tank, through the first stage, to the second stage and out into the water. It does not change the air space between your lungs and your mouth. So if you do have CO2 build up, purging the second stage isn't going to help.
Second, you don't need to forcefully blow air out your nose to clear your mask. In some cases, this wastes a lot of air and doesn't really clear all the water out of your mask. If there is too much force it will cause the water to splash around and not completely clear. You want to breath FULLY into your lungs then slowly BUT steadily blow out your nose to remove the air from the mask.
The best advice I can give you is "it is not a race." It is not a race to the bottom: take your time and keep track of your buddy. It is not a race to get suited up: you don't want to keep everyone waiting but racing to get ready will put you out of breath and a danger to yourself. It is not a race to get into the water. If your buddy is rushing you then you need a new buddy.
In the end, I would suspect you have over-exerting yourself and are not breathing deep enough. I hear people saying that when you breathing on a regulator it should be just like breathing normally. This is a not always good advice. If normal for you is breathing shallow but normally you aren't exerting yourself then you are fine above water. But once you are getting ready for the dive, you are wearing a lot more gear and exerting yourself. You will need to breath more deeply.
In general, you might want to look into breathing exercises. Even if you do breath well and deeply, getting better at breathing will help with your air consumption.
Also remember that scuba diving is not natural. Your body isn't used to this. After you have been diving for a while you will have built up the muscles and breathing to be a good diver. The most important thing is to relax and take your time until your body is used to doing this.