My best guess based on the info provided is>> Offsets are Big Boy Pits and there is a reason they are called Stick Burners you will get more flavor with a pure wood fire. If you look at the overall CI of your Offset compared to your Weber and the flow through the cooker it will explain most of your issue. In the Weber the heat & smoke flow up and around the meat and the area is more compact so your flavor will be stronger VS the Offset where if not designed properly the Heat and Smoke will rise to early and pass over the top and out the stack so it has little contact with you meat Plus when you add the extra Cubic inches of volume it is more dispersed. Think about adding a Baffle and some tuning plates and switch to all wood for heat, some lump in the beginning to help establish a coal bed is fine and preheated splits to maintain your Zone.
I never met a Cow that I didn't like with a little Salt & Pepper! Certified PORK-A-HOLIC