Is my action correct, or does it contain sin, because I did not give the charity to the first orphan I intended it for, but rather gave it to another orphan?

If the giver of charity does not specify a particular orphan, then there is no sin upon you in giving the charity to another orphan, because the charity was given according to the condition of the giver. However, if the giver of charity specifies a particular orphan by name, then you are not absolved of your responsibility by giving the charity to someone else, unless the agent is aware of that and approves it.