Not necessarily Brent---but, in this case, you DID leave yourself open to a fine I think---that is if your state has the same or similar game laws as the two states I have worked in.
All anyone has to do in most states is call the "Warden" BEFORE moving the animal in question, tell him about the situation and that you would like to take the animal.
He then will give you the OK.
He MAY, not always, come out and check that everything is on the up and up and things are as you say or he may just say "take it" I got your info and if another officer stops you, he/she can call me.
You left yourself open for a fine by just taking the animal --again, if your states laws are similar to where I have worked.
If you would have got cited, well, you can only blame yourself.
My answer to OP would be to leave it in most cases.
If for example it was hit by a car and there were witnesses around and an animal was suffering at the side of the road and there was a damaged car there and I had a firearm, I would put it out of it's misery.
If I saw a wounded or hurt animal in the woods, I would leave it alone---if you kill it by firearm in a technically illegal fashion, you are liable.
Dipper