A lot of people don't know that allergic reactions can present as things like this for sure. People almost always jump to anaphylaxis, but even mild lactose intolerance can present as symptoms like that or a simple and short stomach ache etc.

Yes, and there's also the fact caffeine is a diuretic and can cause dehydration if you don't drink additional water to compensate.

Combine that with excess mucus likely created due to the reaction to dairy and you've got a runny nose that's no longer as runny but nasal passages that are just as clogged.