Let it out, don't hold it in. Close your eyes, focus on your intense emotions, breath in and let it out, gently and fully.
I try to do this with all intense emotions be it overwhelming sadness, fury, despair. Feels better than holding back or forcing it out.
Or, in so many words: Cry!
Let it all out, there's nothing to worry about. Just... cry.
I thought I was the only one.. yes. Up - it said more emotionally in those first 10 moments with no dialog than most movies do in 2 hours.
The recent Jungle Book was similar, my eyes were misty the whole movie. The Martian had similar effects.
I hardly ever cry otherwise.
The first ten minutes made me so upset that I spent the rest of the movie fighting my body so that I wouldn't start bawling. It would have been such an intense and loud cry that I absolutely would have had to run out of the movie theater. After the movie I drove my friend home (~30 min), but I couldn't say one word to her because I was still so deeply upset that my lips were trembling.
I don't cry much, in fact, I really can't cry when most people ordinarily would. But in the rare circumstances where something can make me cry (whether it's very sad, or just incredibly beautiful) it is always with maximum intensity.