I feel like the worst mommy in the world

We just spent 2 hours at the walk in clinic (at bedtime) because Trevor sprained his elbow, or I sprained it for him. We were walking across the parking lot at London Drugs, and he tried to pull his hand out of mine so he could run, so I held on tighter, and he tripped and fell, while I was holding tightly to his hand. He whined, and I thought nothing of it, because I thought he was mad that I was still holding his hand. He walked inside, I got a buggy, and then he wanted "uppies", and threw a fit when I put him in the buggy instead of carrying him. I gave him a snack, and he was still sniffly, again, I thought it was because he wanted to be carried. We walked through the mall, were in Zellers, when he's still whining, looks at me and patting his right arm, says, "wowies mommy" meaning owies. He said it a few times, when I realized he wasn't using or moving his arm. He wouldn't move it, was only using his left arm. So finally, I decide to go to the clinic, and DH meets me there. For the first hour and a bit in the waiting room he was laughing and playing with another little boy, all the while not using his right arm, or moving it away from his side. He didn't even use it to climb. I tried moving it a bit, and got a snarly "wowies mommy." The doctor figures that the elbow had popped out a little, and settled back in on it's own, and now it's like he sprained it, and it's a little tender.
I feel horrible. I know I didn't let go so he didn't get hit by a car, but I still feel horrible. The doctor and DH both said I did the right thing, but it doesn't feel good when you have a hand in your child getting hurt. Let's just say the tears flowed. Let's hope that he wakes up tomorrow feeling better.