I've received quite a few responses and I would like to thank everyone for their input. I've come to accept the multimeter I bought is terrible. Thank you RadioShack. I should have realized that if it doesn't cost much, it's probably because it's not worth much. I recommend it to no one.
With that aside, I added the .01uF capacitors between the ground and Vcc pins as close to the chip as possible per dc42's request (and in Nick's page).
Nick, the socket on the right you couldn't read in the picture says V / ohms / mA. However, I moved to the left socket as suggested and I am seeing a reading that jumps between .002 and .003 Amps (2 - 3 mA). That's the highest resolution I can get from the socket. This reading is consistent with what I had from the borrowed (non RadioShack) meter and is also consistent with the ATMEGA328p datasheet as pointed out by user dc42.
Regarding the note on the ATMEGA328p's reliability under 2.8V @ 8MHz, I will keep this in mind. I tested with a 3V CR2032 and saw the roughly the same results for current. I also loaded a sketch to blink pin 13 and I didn't see any inconsistent blinking to indicate it is rebooting. The project this will be used in will run @ 3.3V, I just don't have an unregulated 3.3V power source at the moment.
Overall, I'm quite happy to see the current drop from about 16 mA to just under 3 mA, but I do see a conflict between what the ATMEGA328p datasheet states and what Nick's test is showing me.
I'm curious to see what other people are reading for current.