New Thermostat installed but compressor comes on immediately
Ultimately I figured out what I was doing wrong with help of this question: 
Adding a C Wire To A New Honeywell Wifi Thermostat

I was mistaking the 24VAC transformer for the relay to the outdoor unit. The only thing I really needed to do to the original wiring was connect the blue wire ("C") going up to the t-stat to the splice bundle near the furnace.