What is the make/model of that camcorder?
When you Open one of those AVI files, in the great, free utility, G-SPOT, what does it say about the file? Posting a screen-cap of the G-Spot screen will be very helpful.
This ARTICLE will give you some background on AVI's, which are wrappers.
The camera saves its video in .AVI format
There are no standardized formats that save HD in an AVI wrapper. The likely explanation for this issue is the non-standard format of your camera.
One solution is to use a consumer level editor. They're often more forgiving when using consumer formats. Another is to use a camera that has either the HDV or AVCHD logo right on the camera. This way you will be sure to get compatible media.
There have been some noted problems with AAC Audio. As a test, I would rip the Audio in Soundbooth, Audition or the free Audacity. Do a Save_As choosing PCM/WAV 48KHz 16-bit, and Import that. Drag it to a vacant Audio Track. If you have the CTI (Current Time Indicator) at the beginning of the Video Clip, and Snap ON, it will Snap into place. Does that work?
The problem is not the audio. The audio ports into Premiere just fine. There's just no video. It's sounding like converting the file into another format is the only answer. I just don't want to lose any quality. Is that possible? What tool would you suggest? I can't use Media Encoder because it has the same problem--it only recognizes the audio. For future people with this problem, I thought I had better actually include the name of the camera in the text of this thread. It is a Toshiba Camileo H30 1080p HD Camcorder. Actually, the camera came with a program called MediaConverter 3.0. It's such a dumbed-down program though and has very little customizable options and is taking forever to "convert" one file. It's frustrating that it appears I'm going to have to do this with every clip I ever record on the camera...GRR.
Tools to Convert to DV-AVI http://forums.adobe.com/thread/415317?tstart=0
Convert your files to DV-AVI Type II with 48KHz 16-bit Audio
I have NOT used the products below, I only forward due to other mentions
$90 http://www.magix.com/us/movie-edit-pro/ plus $5 Ship
$70 http://www.nchsoftware.com/prism/index.html Converter
$40 http://www.deskshare.com/dmc.aspx Digital Media Converter
$00 http://www.dvddecrypter.org.uk/ or http://www.mrbass.org/dvdrip/
$00 http://www.squared5.com/ MPEG Streamclip Converter
$00 http://www.erightsoft.com/SUPER.html Multi-Converter <-- supposed to be very good
$00 http://www.virtualdub.org/ Mpeg to AVI Converter
Yes right. I focused on the wrong stream - sorry about that.