The IC regulator attempts to keep the reference voltage (between the Vout and ADJ pins) at 1.25V.
This is different than what I thought. I thought the IC regulator attempted to keep the reference pin voltage AT 1.2V
(from ground). In that case, to regulate to 27.5V you would want a ratio of R1/R2 of about 26.3/1.2=22. So my bigger R1 makes sense.
If the regulator attempts to keep the reference voltage at Vout-1.2, then you would want a ratio R1/R2 = 1.2/26.3=.045. So it looks like I need to switch my pot and resistor around (and maybe use a bigger resistor as well).