First off, it's probably unlikely that the deer you shot is dead. Though not 100% sure, from the description you gave of the blood you found, it sounds like you might have hit bone. Also, judging from that fact that you didn't find "initial" blood in the area where he was standing when you shot him likely means the shot you put on him was not critical. Usually when you have to walk 300 yards to find 1 spot of blood, it usually means the deer is not hit critically at all. There have been instances where a bullet can actually "seal off" the flow of blood for some amount of yardage until it breaks free of becomes dislodged and hunters will start to find "good blood" several yards into their tracking. But usually if this was to happen the deer would start pushing out blood at a good rate and usually far before 300 yds. Watery blood is usually a distinct sign that you likely hit some bone somewhere on the deer's body.

I do have a couple questions to ask you, if you can think back and recall your observations when you watched the deer trot away after the shot.
1) Was he limping at all when he was walking away? Although a shot to the front shoulder is not always critical, a deer will usually limp when it is hit in the front shoulders.
2) Do you recall any hair being found in the area where he was standing when you shot? The color and even length of hair found in that area can also indicate where you might have hit him. White can mean belly hair, throat patch or even rump areas, where as whitish beige color hair can mean a grazing behind the legs anywhere from the " elbow knuckle area" up to the base of the brisket to where the two intersect. And obviously dark brown hair would be a sign that you hit him up top near the shoulder area.
3) If you didn't get to see if he was limping away, did you at least get to decipher some really good track impressions when you where following his trail? Was there anything unusual about any of his tracks, like finding only (3) tracks making contact with the ground instead of the usual 4? Where there drag marks on any of the tracks that would indicate he was favoring one of his front legs?

Now, on to deer walking in circles:
I don't know if you have ever read any books on tracking or stalking deer by following their tracks. But in a whole bunch of the stories a lot of hunters have made observations that deer usually will walk in large circles, even if they are being followed by a hunter. And what's even stranger than that, some stories say that they will walk in their circular patterns even if they KNOW that a hunter is on their tracks following them. If you ever get the opportunity to follow some tracks and stalk a deer, you are going to learn a whole lot about deer. (especially when they know you are hot on their tail) They can take you on a pretty wild ride and they have a lot of tricks up their sleeves when they know they are being followed.