anxious care: the weight of worry a person carries, either over their own life or on behalf of people they are answerable for
This is worry with weight to it, not a passing thought. Every one of the six places it appears treats it as a load: something that presses down, takes up room, or has to be lifted off. It is the noun beside the verb "to worry" (μεριμνάω), so where the verb is the act of fretting or caring, this word is the burden itself.
Four of the six sit in one picture, the cares of the present age. In the parable of the sower these cares grow up around the message like weeds and choke it, alongside the pull of money and pleasure, and Jesus warns elsewhere that the same cares can weigh a heart down until a day arrives that no one saw coming. What the picture adds to the plain idea of worry is competition: these concerns are rivals for the room a person's life gives to other things.
1 Peter turns the load around. Because it is a weight and not a permanent feature of a person, it can be thrown off, and the readers are told to throw the whole of it onto God, on the ground that he himself is concerned about them (a gentler word for concern, μέλω). The command assumes the anxiety is real and heavy; the remedy is to move it, not to deny it.
Paul then shows the word does not have to name a fault. Listing what he suffers, he ends with the daily pressure on him, his anxious concern for all the churches. That burden is aimed at other people's welfare rather than his own comfort, and he offers it as evidence of faithfulness, not as something to repent of. So three different things happen to this one load in the New Testament:
- it can choke a life until nothing comes to fruit
- it can be handed over to a God who is concerned about the person carrying it
- it can be taken up willingly on behalf of others