Adapter for drum module showing incorrect values?
DMM's do not measure RMS but will display RMS for a sinewave  ( Unless otherwise labeled)

This is done  by measuring some other method like dual-slope integrated average or full-wave rectified Peak, then converted to RMS by some ratio for a sine wave.

Judging by your measurements for a 9V supply, of 13.4V I suspect it is an unregulated supply, which uses a full wave bridge to Caps and is thus 1.414 x the RMS voltage with no load.   There is also transformer conduction losses so that at full load the voltage may will drop to 9V with a resistive 1A load.  This means it may rise another 5 ~ 10% with no load resulting in 1.5x9V=13.5V.

When the cap is charging just before the peak, to supply a load, the duty cycle may be say 10% and thus 10x the current needed to supply the DC out.  THus if your DMM measured Peak voltage or current and converted to RMS using 0.707 for a sine wave, then you may be actually measuring  peak current of 4.3A/0.707 = 6.1 A but at a duty cycle ~ 10%~15% so the actual average current may be 0.6 to 0.9A amp