You may want to measure (or calculate an estimate for) amps for the sake of
(1) buying a large enough power supply or debugging
(2) checking the ampacity of your wiring
(3) checking the voltage drop of your wiring
If instead you are concerned about input power and your power bills, I agree that you would do better using one of the AC mains power meters like the Kil-o-watt - this can factor in the real power supply efficiency.
These are two different things, tho.
Also, regarding "varying current" this happens in two way on two timescales. When you are showing an individual pixel at 20% of max brightness (eg: at intensity 51 of a 0-255 scale) the chip next to the LED inside the pixel housing will be turning the LED on and off many times per second, perhaps hundreds so that you don't see flicker, so it can keep it on about 20% of the time. The high frequency is so you don't perceive it as flickering. The overall current draw will be a combination of these many individual and unsyncrhonized PWM current draws, so it'll be pretty jumpy on a millisecond time scale. Your DMM could be sampling that value at various points and not averaging it well, so it may have unstable or inaccurate readings. The analog meter (and maybe GOOD DMMs) will be more stable.
And of course, there is a different variance in that you will be turning pixels to various brightness levels during your sequences. Hopefully you realize that you would not want to be varying the lights while you are measuring them with the DMM, tho. This is the varying timescale that Nutcracker (or Vixen for that matter) could estimate for any given sequence step.
So there's input AC input power to pay for vs DC current to plan wiring, and there's PWM noise to confuse DMMs vs sequence controlled brightness changes. All are valid things to consider, but different.
