What you described is a classic ground issue. The problem is that the added amperage draw of the head/running lights being on for your vehicle makes the trailer taillights a bit weaker.
What you would want to is run the grounds of the light assemblies of your trailer back to the main ground harness of your trailer. I can almost guarantee that will fix the problem.
I attached a help article for you to check out as well that covers 4-way troubleshooting. view full answer...
The Curt harness # C59236 you purchased is the correct one for the 2009 Audi A4 sedan and the odd behavior from the trailer lights, like the turn signals activating on the opposite side, is most often caused by an insufficient ground. This usually has a very simple fix.
Remove the ground wire ring terminal and apply sandpaper to that point to remove any paint, primer, undercoat (or rust) that could be blocking the clean ground connection. An application of dielectric grease like # 11755... view full answer...
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