This trait you speak of was raised in a therapy session and the idea is that as a child you maybe took on some responsibility that should have been done by an adult and so as an adult now you feel the need to help everyone and care about everything because, as a child, that's what you were asked to step up and do.
My therapist said that it really is not that you have to stop caring, but it's, as you said, have to realize what you can do and what is out of your control. What is your responsibility and in what situations other people should do some lifting, in other words..
For example if you see a dog by the side of the road it isn't your responsibility to pick up the dog, take him home, bathe him, give him food, etc...but you don't just have to say "well I don't care, not my problem" and drive on either..
Someone like us (I suffer from this too) simply should pick up our phone and dial a shelter or someone to come get the dog.
May not be the best example but I think it illustrates our desire to do it all when there are other people who can help us and/or things that are not our responsibility don't have to be a burden that we assign to ourselves, simply because as kids we had to, we don't have to as adults in each and every situation in which we could "be the hero"
And in fact, if you look it up it's called A Hero Complex in regards to childhood neglect of some fashion. People like us basically have to learn to trust others to help take care of the world around us and not try to do everything, because we can't do everything and when we try we fail, then failure gives us anxiety.