The errand taken over completely gives more relief than help that still needs constant instructions.

Help that has to be supervised is not entirely help. Explaining, checking and correcting take their own energy, and for someone already stretched that can cost nearly as much as doing the task alone. Taking something over completely, including the decisions inside it, is what actually removes weight.

When you offer help this week, try offering a whole task rather than your availability. Say what you will do and when it will be done, then do it without going back for approval on every small point.