Hi pinot, welcome to the forums.
avast doesn't stop outbound or inbound traffic (that's a firewall task). A lot depends on how your ads are downloaded, but I assume that they are through the usuall http port 80
So the only thing that avast would do is check for infection in your temporary internet folder, etc.
If they are pulled down some other way through your email server (which I can't see how this can be done), port 25 then there is the possibility that the email account settings need to be checked.
Try the
Users FAQ - Mail Protection to confirm your settings are ok.
Let us know if you are still having problems... there are many people using eudora, not sure if they are using the sponsored version though. There is a later version than 6.1 now.
HTH David