My contact person at Lavasoft has read your logs and she can see that the uninstallation failed and parts of Ad-Aware wasn't uninstalled. Those parts disturb a new installation.
The attached fix will bring back Ad-Aware to Control Panel and you can uninstall it. Before installing Ad-Aware again, please let me check that everything of Ad-Aware is gone by attaching a new DDS.txt.
Please download the attached file and save it on your desktop.
Right-click it and select to extract its file.
Move the extracted AdAwareUpdater_win32_11.1.5152.0.msi to the folder C:\.
Start a command prompt:
Start menu - All programs - Accessories - Command Prompt
Enter these commands (end each one with Enter):
msiexec /I AdAwareUpdater_win32_11.1.5152.0.msi INSTALLDIR_ROOT="c:\\Program Files\Lavasoft\Ad-Aware Antivirus"
Please, remember that programs as RegistryBooster and DriverTuner can destroy things.
The attached file and the instructions are only for 32 bit XP. If used with other Windows, it might destroy something.