Actually I had this problem too (so I stopped using shareaza :P). What sometimes would help is clearing the “incoming downloads” folders (I forget the exact name) where it stores semi-downloaded files, incase it is filled with stuff you later cancelled downloading.
In the Standard Shield’s settings, turn on the “Show detailed info on performed action” option (it’s on the Advanced page) and then try to download some file & close Shareaza.
That way you can see what file(s) avast is scanning (there should be a popup window displayed in the bottom-right corner of the window showing this information). If those are not irrelevant files you could exclude them from future scanning.