Connecting Common Wire from thermostat to Air Handler
For systems with two transformers, you would normally have a red wire for heat and a separate for cooling (commonly labeled Rh and Rc on thermostats). So, it seems like you just have one transformer.

Did you measure 24v between the red and brown wire exiting the air handler? Also, common is commonly attached to the metal body of the air handler, so you can also check for continuity between brown and a good metal "ground".

Great diagrams!  They make answering a question like this much easier.