There has been a couple more Adaware updates since.
Download & instal Spybot S&D from http://www.safer-networking.org/index.php?page=download Update it B4 scanning.
After the scan is complete, have spybot fix everything marked RED.
On the page that first opens when you start Spybot there is an option to immunise, you should do this. In the immunise section there is also a link to download Spywareblaster. Download that & you can keep it updated by selecting the same link that you use to download it.
Download CWShredder from http://209.133.47.200/~merijn/files/CWShredder.exe & run it. Select the fix button & it will get rid of everything related to CoolWebSearch. Close ALL other programs including IE before running CWShredder.
Stay with these programs, they are free & better than spysweeper. If these fail (after running them in safe mode(not that you have to)) then post that HJT log.