A favor offered directly can be easier to accept than an open-ended promise of anything.

Anything you need is a generous sentence that is almost impossible to use. It hands the work of deciding, choosing and asking back to the person who is already overloaded, and most people respond by saying they are fine because that is simpler than composing a request. A named offer removes all of that.

Instead of telling someone to let you know, say what you will do and when. I am bringing dinner on Thursday, or I can collect the children on Friday. Give them the easy option of saying no thanks, and make the yes require nothing more than a nod.