I purchased the Arma 3 DLC but I still see the ads for it.

Please try to do the following:
1. Exit Steam.
2. Go into the Steam installation folder and find the folder called "appcache".
3. Delete the "appinfo.vdf" and "packageinfo.vdf" files.
4. Right-click on Steam icon.
5. Left-click on "Run As Administrator".
6. Right-click on the game in Steam Library.
7. Click on "Properties".
8. Select "Local files" tab and click on "Verify integrity of game cache files" button.
9. Please wait, the process can take several minutes.