You did not mention having investigated the warning light sufficiently to have an engine code. Get your VCDS out and see what the engine is telling you.
Having recently done major engine surgery and having fresh recollections of the systems, I have thought about this and have some conclusions. In my opinion, to have coolant get into the intake or exhaust system from the EGR such that it would get into either the intake or exhaust would require a leak internal to the cooler itself, which is made of stainless steel. As far as I am aware, these coolers do not corrode such as to leak coolant into the exhaust gas chamber.
It is quite possible however, that the observed symptoms are actually unrelated, rather, simply coincidental. I suggest your examining the heater core hoses, as it is common for them to degrade and spring leaks due to engine oil getting on them from the charge air system or turbo leaking small amounts.
The simplest answer is probably the correct one.