To be very busy doing many different things.
run around
have one's fingers in many pies
To be involved in many, or too many, different things.
otherwise engaged
busy doing something else
up to one's neck in alligators
Extremely busy; having many pressing obligations, distractions, etc., …
sugar and spice
Two things that are fundamentally different from each other.
chalk and cheese
Said of things that are very different, though possibly superficially …