Impact: Last year I shot a buck with a 270 (150 gr) and it reacted much like your first one. I was so surprised that it didn't just drop in its tracks that I didn't even attempt a second shot running in the woods. I could have in hind sight. I sat for about 20 minutes and walked over to where I should have hit the deer. Finally found blood. I slowly began following the blood trail in the dry leaves marking the trail as I went with tiny pieces of surveyors flagging. I didn't know if the deer would be 75 yds away or run for 200 yds. So, I marked the trail. Found the deer out of my sight in the brush at about 75 yards from where I shot. I hit the deer just a tad back from the front shoulder, and the bullet just wizzed through without much expansion. Inside, there was lots of lung damage. I try for shoulder shots in order to try to break the front shoulders and take out the lungs/heart at the same time.
This year, I shot one with a handgun (480 Ruger with 325 gr HP) and it ran about 50 yds. Again just out of my area of visibility. Broke the front shoulder. Took the same careful precautions and found the deer in about 30 minutes after he bled out. There was lots of blood as I followed the trail, unlike the 270 deer that just had spots of blood in the leaves but a steady trickle. Both were shot from the exact stand location in a similar impact location at about 50 yds.
I swear by the use of flagging to mark a trail as you just never know when the blood will stop and you need to be able to sight down the trail to give an approximate bearing. First the bearing, then the widening circles in search of the deer or more blood. If I see the deer lying down, I check to see if it is in a "natural position" and watch for any movement/alertness. I would take another shot if I saw any activity at all.