How to connect faces without intersection? I don't want to change the geometry of the faces I want to connect to. I'm sure more seasoned 3D modellers have a simple way for this. Please enlighten!

In the tutorials for joining cup handles to the cup they line up so it's not an issue. Another tutorial even extruded the start and end of the handle and then joined them in the middle. I don't think either would work here.

Here is the cross section:
You can use Boolean modifier to join them together. If the handle is in the same mesh of the cup body, just simply separate it as a new object by P, then boolean the two object. You will get a final shape without intersection after apply it.



NOTE: When using Boolean modifier, the intersection part has to be closed shape.