I teach elementary art and have no sink in my classroom. This is what I do.
I assign a student to each of the following tasks:
Pass out newpaper
Pass out water containers and paintbrushes
Pass out papers
They cover their tables with newspaper.
They get paint shirts by table.
I give them paint in egg cartons.
At the end of class, they put the paintings in the drying rack by table.
The water container people take the container and paintbrushes to the
nearest sink to wash them out.
The egg cartons with paint go in the trash can (yes, it does waste a little
paint, but saves on clean up)
The newspaper people clean the newspaper off the table.
Handwashing is the classroom teachers responsibility. I can't watch them in
the bathrooms sufficiently, and they all have sinks in their rooms.