An update, for anyone still having problems.
Having failed (through inefficiency) to take my copy of AdAware SE 1.05 home to install on my computer there, I thought last night that I'd have try once more to update the files for my AdAware 6 that's already installed on that machine.
It worked, for the first time since this problem first cropped up. So it may be that the problem has been sorted out at Lavasoft's end now.
I still need to take the new software home to install it, though - downloading it on a dialup take so long.