I guess I was not paying attention.
So... What kind of computer? The DCR-HC38 is miniDV tape based. Link to the camcorder manual:
My normal import method for any type of miniDV tape based camcorder would be via firewire (AKA: IEEE1394, DV & i.Link). While most computers today don't have a firewire port, many have a way to add on (and many don't). If your computer has an available PCMCIA (desktop/tower) or PC Card slot (usually older laptops), the adding a firewire port is do-able. If your computer has an available ThunderBolt port, there is a Firewire adapter. In either case, this bypasses the external analog/digital converter (EasyCap).
There is no USB-to-Firewire converter/adapter that will work. Once upon a time there was a way to use USB (camcorder-to-computer) for the video transfer, but the USB driver has not been supported/updated since about Windows XP, so don't bother trying to get that to work on any Windows OS newer than that.
This does not resolve the question as to why the audio gain is so high - and I am still thinking about that. (I use an Elgato external "analog/digital converter" for the occasional VHS tape conversion, and have not had the audio issue you describe).
Reading through the manual, page 22 references adjusting the playback volume (though it does not specify if this is playback volume of the camcorder when viewed on the camcorder or the output to the AV-out (which, in your case, connects to the EasyCap).
And... another possible option... If your computer has an audio input, you could connect the left/right audio-in from the camcorder (presumably with a 3.5mm to left/right RCA cable tail) and capture the audio separately and directly into Audacity (bypass EasyCap)... Then add the audio to the EasyCap captured video and mute the audio that is "too hot" (assuming the video editor you are using has that sort of audio manipulation capability). The reason to initially keep the "too hot" audio is to use it to sync the good audio. When the echo is gone, the audio and video are in sync... But this becomes as cumbersome as the multi-step DVD recorder process you currently use... or you could get the audio off the CD and add that to the EasyCap captured video and mute the EasyCap too-hot audio...
I'm still trying to figure out why the (acceptable) audio goes icky during the EasyCap transfer...