You may have missed the biggest factor I stress when leaving the hide on (or not), the only place to store a hanging deer is a controlled temperature cooler.
During our processing rush we will have deer with the hide on in our cooler at a tempeature range of 29 - 34 degrees for up to two weeks with no problems as long as they were fresh when the went it.
As far as hair on the meat, my guess is that you possibly skin with the head up, and may being doing to much cutting with the knife and not enough pulling with your hands.
We skin with the head down, and make sure that any cuts that are made are done from under the hide with the tip of the knife and then
brought up. As you work it loose from the hindquarters minimize the knife and pull with your hands as much as possible. I have found that instead of pulling straight down, work the meat Loose along the flank and then pull it down and back towards the top of the back. The hide should come off clean with little meat attached.
As soon as we're done skinning we spray the carcass with cold water to remove the little bit of hair the may be on it. While cutting we keep a bone scraper handy to remove any hair that was missed. You may be able to buy one through you local supermarket.
Dave