Well, a few thoughts...
First, I disagree slightly with the guy in the video. Yes, put the mic on a boom but you want it to be slightly in front of you and aimed roughly at your mouth, not your chest. Even with a short shotgun like the NTG1, the closer the better. (Well, not SM58 close but every time you double the distance from your sound source, your levels become a quarter what they were before. You don't mention whether you're shooting video with this or just audio recording but get your mic in as close as circumstances allow. Also, depending on your layout, sometimes it can work to boom out of shot below your face rather than above.
Second, your levels aren't as bad as you think for an original recording. As has been explained many a time on these forums, there are two different standards for the metering of a recording. Ordinary analogue recorders/meters use dBu or dB(VU) where the zero level which is determined by an arbitrary voltage. In that system, you can allow peaks in your recording go above this zero mark, by how much depends on the quality of your gear but between +4 and +8 is fairly normal, allowing lots of overhead for safety. On your digital gear though, they use dB(FS) which stands for "Full Scale". Zero level on the Zoom or on any meters in Audacity is the absolute maximum you can record without clipping distortion.
The norm is to say that 0dB(VU) is equal to -18dB(FS). This means that your -6 level is, in fact, like recording at +12 in analogue which is actually a very hot signal. Normally you want to aim for the "average" of your waveform to be around -18 with peaks up to -10 or so. Trouble is, DAW waveform displays make this look really tiny--you have to go by the meters not what the waveform looks like.
Once you've done your recording, you can then (sometimes) boost these levels depending on what you're doing with the recording. Ending up on a CD, it's pretty common to boost things up to 0dB(FS) (using a limiter to avoid going above that level). For some broadcast purposes, you work to a standard specified by the broadcaster...for example our satellite transmissions, we demanded that audio be limited to +8 (i.e. -10 dB(FS)) to avoid cross talk between satellite channels.
Just as an FYI, I've used an NT1G in the past--I used to carry it as a back up to the Sennheiser 416 which was our primary mic--and results tended to be very similar between the mics so (barring a faulty unit) that shouldn't be the problem.