"I adjusted the settings on the H2 (recording level & mic gain) so that the level didn't appear to be overloading while I was recording & it sounded OK in headphones. But when I looked at the WAV file with Audacity, the singing was very very quiet and the dynamic range was tiny"

I would expect full scale on the H2 to correspond with full scale in Audacity, but there may be a difference mid-scale - Audacity shows levels on a linear scale by default, but your H2 may well have a logarithmic scale. That means a level at a perfectly usable 20dB below full scale shows as only 1/10 of full scale in Audacity unless you switch Audacity's display to "Waveform(dB)" instead of "Waveform".

If you aim for peaks around 10-12 dB below full scale when recording you'll have a decent overload margin for the odd extra loud peak. When you process it with Audacity you can use the volume control "effect" to boost it. You should aim for no peaks higher than -0.5 or -1dB below full scale (because the DAC on playback can overshoot), and if it requires something like 3-8 dB gain to achieve that you've got it about right.

If you are expecting to do a lot of processing, especially EQ and compression, you might find you can record at 24bits and import at that level into Audacity (I think Audacity uses 32bits internally) but quite honestly that level of perfection is wasted on something that started life as a home recording and certainly if it's going to end up degraded to MP3.