You're getting an adrenaline rush because you're being injected with adrenaline
The compound that slows the bleeding is adrenaline. Most people don't notice it, others are more sensitive to it, so I'm guessing you're one of the latter...
Tell you a slightly amusing anecdote, about a million years ago, I was working in the Oral Medicine department of a big hospital. My next patient was coming in to get the results of the biopsy that one of my more junior colleagues had done 2 weeks prior.
No big deal, it was a routine biopsy for something fairly trivial.
However, the patient came in looking like he was about to face a firing squad, complete with super anxious wife holding his hand like she wouldn't let it go.
Before I could open my mouth, he blurted out "Well doc, how long have I got?"
Blank look from me...
"Before the cancer gets me?"
Again blank look... "What cancer?"
"The one I got biopsied 2 weeks ago"
I honestly nearly laughed out loud before realising how upset the poor guy was, when I told him it was nothing to do with cancer he about jumped out his seat and started hugging me. So did the wife.
My young colleague was more interested in getting a good sample for the lab and hadn't really explained to the patient what was going on. Poor guy had been re-writing his will, sorting out his life insurance and goodness knows what else.