Hi pwollner,
I feel some of these replies are a little over-done.
You might like to consider something simpler.
I haven't worked out the resistor values,
they depend on the SCR.
The two diodes would be chosen to give appropriate
voltage to the led.
The resistor on the gate would be chosen so that one
drawer would not engage the SCR, but more would.
Any drawer would light the led,
any two (or more) would operate the buzzer,
the intermittent nature of the buzzer would turn it
off when the drawers are closed, or one left open.
SCR's have a very high resistance, almost open circuit
when not operated, not enough to affect battery life.
Two or maybe three U2 type batteries should give a
very long life, its hard to guess but i would expect
a year, more maybe, depends on usage.
Sorry this took so long, i think i had a mental
block.
Best of luck with it, Regards, John