I use Avast Internet Security Suite and it works great for me. I do not use AdAware so this post is bit off topic.
However, I also use Malwarebytes and SAS, which were mentioned above, once a week as a double check. They very rarely find anything.
However I also use iObit360 (free version) once a week and it does occasionally find cookies and even trojans that Avast missed. Afterthought: This program gets a lot of bad user reviews. I have used it since it first came out and while version 1.0 was buggier than all get out, the current version does want I want; i.e. occasionally finds cookies and trojans that other programs don't. Just make sure it doesn't stay in memory when you are not using it.
P.S. None of these programs, i.e., Malwarebytes, SAS and iObit360 should be kept in memory when not updating, scanning or handling any files it might find on the scan. When you done using it as a batch program kill any program related Process that remains running even after the program is closed. The Process can be killed with Windows Task manager. I have never had a problem killing any of "these" programs "Processes" with Windows Task Manager. If the "Processes" remain after you close the program you are unnecessarily using resources.
Finally it is best to close everything else when running these programs though not necessary.
Let Avsst do its thing. It works great but like every other software program in the world it is not perfect. So doing a once a week or once every couple of weeks double check with these other programs doesn't hurt.