Congratulations. Folk spends lots of time and money on the video gear and forget about the audio.
Poor video can usually be salvaged. Poor audio is often terminal.
I use a beachtek DXA-5Da with my 7D, T3i and T2i. This gives me the necessary AGC disable (not an issue on the 5D3 with it's manual levels) but also 2x XLR inputs and a headphone socket. Again you don't need this for a 5D3 as you can headphone monitor straight from the camera.
So you could actually go for a more basic and cheaper beachtek that would give you synched audio within the video files off the memory card.
They do a model for camcorder users, the DXA-2T, which would suffice for your camera (as the 5D3 has a headphone socket built in) with the right microphone.
You'll need to cut the video pin off it, but apart from that it's perfect for a 5D3.
I'm not familiar with the azden brand. I had a google and to be honest I would consider changing it.
I use a Rode NTG-2 for my home kit, and a Sennheiser K6-ME66 combo for my work kit, big price difference, but the cheaper rode is actually a very very good mic for not a lot of money. Both these mics take an internal AA battery, negating the need for a phantom power source, and run quite 'hot' keeping the camera pre-amp levels low, so the cheaper DXA-2T would suffice.
Other benefits to the Beachteks:
mic line switching
independant channel levels (basically attenuators, use full up unless level really hot)
stereo (independent L & R tracks) and mono (linear stereo, basically both mono inputs mixed together on each of the audio tracks)
If you get the DXA-2T (without phantom) there is no power required, so nothing to charge up or forget to power on)
It fits on the camera tripod thread, so it can be left on the camera (no rig or gaffer tape, mounting gubbins required) and you can still use the tripod with it in place.
Don't get me wrong, in terms of absolute sound quality the zoom is a better file format, better preamp etc, but a whole lot more hassle. My set up is as good as what I was getting on my ENG cameras.
I use beachteks and I'm doing this professionally. If the sound was poor I simply wouldn't use it.
Get yourself a set of half decent closed back cans (AKG-k450s are good budget monitors, sennheiser HD-25-II's are the default ENG headphone, very very good but triple the cost)
I would also consider getting a mic stand for interviews, i use a beyer dynamic having went through about five cheap ones. Mic stand is brilliant for micing off camera out of shot.
The very cheapest option for your synching problem would be to run a short minjack lead from the zooms headphone port into your body. But keep the zoom headphone levels very low and use the manual record level in the 5D3.
This to me is using a sledge hammer to crack a nut, and still gives you problems of charging and mounting, etc.
link to beachtek stuff:http://www.beachtek.com
If you really want a unit that provides phantom then look for a used beecjtek DXA-4P, which takes an internal 9v battery and provides phantom.