Sorting an unordered pile of items into drawers with minimal drawer movements
How about the following:


Go through your bag; fold and stack items for drawers $D_i$ and 
$D_{i+1}$, $i \in 2\mathbb{N}$, in/on drawer $D_i$ 
(which you open on demand).

Make sure you have the items for $D_{i+1}$ on top; that may require quite
some reordering pancake-style, 
but you don't seem concerned about such operations.

Note that there is always enough room (two drawer heights) to do this,
assuming the drawers are roughly equal and your clothes fit into them as it is.
For each open drawer $D_i$,


pick up the stack for $D_{i+1}$ (and put in the bag, if necessary),
close $D_i$,
open $D_{i+1}$,
put the stack in there, and
close $D_{i+1}$.



This algorithm requires $2M$ drawer movements if all drawers get clothing, so it is worst-case optimal.

It will waste some movements if there are items for $D_{i+1}$ but not $D_i$; if only odd drawers get items, we execute twice the drawer movements necessary. Since this is the worst that can happen, we still have a $2$-approximation.