Tailteann, I've afraid I don't have any magic answer for you, just some thoughts.
I don't know that you'd have to dump Avast. The AVs usually don't damage each other and only fight when you have more than one resident scanner on.
Obviously, if you can isolate whether it's a browser or add-on, or whether it's Windows itself that's hooked (probably browser level, though), you'll have a better idea how to procede. If Windows, you can do the obvious things to try and find nasty little suprises. You've probably already checked your autostart programs for strangers, don't forget to check windows services, too. Some hackers try and hide them there. In any case, you can also use search to search by date. You can search "date created" by a specific day and maybe you'll find a suspicious file.
I assume this is happening with No Scripts on (I LIKE that add-on, but it definately takes some getting used to). If not, you might want to try to put up with it blocking all sites initially and see if it still happens. That would obviously indicate that it's a script, so at least you'd know what you were looking for. I assume you checked for a new add-on, more importantly, have you tried disabling any add-ons that might be vulnerable, or all of them - you never know. I doubt you have any, but something like iMacros or even a an autoform utility like Lazurus could be vulnerable. Then you can try uninstalling-reinstalling that add-on, or just do without.
Have you thought of reinstalling FF. Or trying IE. I'm saying trying IE as a test to see if you have the same problem. If you don't, that obviously would hint that only FF is hooked and would hint that a reinstall might help - that the hook was in FF, itself, and might be broken by an uninstall-reinstall. Do you have FF version 3.5? If not, that would give you an excuse to do a new install, anyway. Though I'd do a full uninstall of the earlier version, first. And I'd do it with revo uninstaller. Have you used that? http://www.revouninstaller.com Revo tries to read the program's structure - it may just read the install or uninstall logs, or it may try and read the various links (the "calls") to other files - so you might get lucky and it might find a "hook" file if the hacker didn't take the necessary precautions to hide it. I'd use it on it's most aggressive mode. People say that mode is too slow, but I haven't found that. In fact, Revo's good to have, anyway, because it can sometimes uninstall when the built-in uninstaller is corrupt, and it has some other tools, like a junk file cleaner to augment your ccleaner. Besides, it's free and even has a portable version, which seems to me to be just as good as the main version.
As to finding brand new viruses, that's, obviously, a problem. I only used the AVG free version and that was about 4 years ago. I assume it has heuristics to detect suspicious behavior from new viruses and you have that feature turned on. If not, it's probably worth it. I use the AntiVir free version and am pretty happy with it. It's heuristics can be annoying, until the community complains enough about false hits so that Avira fixes them. I've had that happen on my AT&T homepage and on TV Guide's site. Still, it doesn't happen that often, so it's more than worth it to me.
In any case, I don't think you'll have to reinstall windows. At the worst, this is an embarassing annoyance. Take a deep breath and just be patient. You'll find it.
I should add, another way to prevent this in the future is to try a virtual surfing. Such as Sandboxie or Returnil. I use returnil to test GOTDs. http://www.sandboxie.com, http://www.returnilvirtualsystem.com The only problem is that if you forget that you have them on and make a bunch of changes or turn them off and try to access recent surf history, it's all gone. Still, everybody's surfing habits are different and it might be right up your alley.
