Connect one of 16 loads to power (12 V, ~300 mA), decoder IC?
Yes, there are IC's that provide de-muxing as you require, though as mentioned mostly 8/7 bit wide.  

TI has a range:

TPIC6259 provides 8 * 45 V FET outputs with either a demux or addressable latch mode. 

TPIC6C595 provides a serial to parallel SR (less wires, but not a demux) that reaches 8 * 35 V @ 250 mA if you only have only output on at a time. You can series connect as many as you like. 

And of course the FET replacement for the UL2003 type drivers ...the TPL7407 can provide 30 V @600 mA.
Toshiba also have a nice IC that has the diodes for inductive loads built in ...the unfortunately designated TBD62083