I have been told that running water is a no no. Even clean water, I have seen people spray them out with a hose after they hang them up. I was told ( on this forum I think) that the water will cause more problems then it will do good, bacteria issues.
Like everyone else said the sooner the better, I know people who have shot deer and not gone back to look for them for many hours, or even the next morning and the meat is fine. Again if it was 90 degrees outside I would not let one set a full day, but during our gun season in Indiana it is usually pretty cold by mid to late November.
Did you mention, are you gun hunting or bow hunting, when does the season start?
I gut mine as soon as I get to them and hang them in the garage. If it is above 50 degrees I buy a couple large bags of ice and put them directly into the cavity. If it is below 50 degrees I don't mess with the ice. It temps are near or at freezing I will leave one hang for a couple days. Even in the 20's it usually doesn't freeze the carcass in my un insulated garage. One time I hung one in the evening and we had an unexpected cold snap come through, I had to work the next day and the highs were only in the teens. By the time I got off of work that next night the deer was solid as a rock. A pain in the A$$ to skin and I ended up using a battery powered "sawz all" and cut the entire thing into sausage meat. I will never let one freeze solid again.