I definitely relate to this. Even when a happy moment happens I tear up a bit and if a sad one comes on I have to really fight to not do more than slightly tear up.

I wasn't always that way though. I used to show emotion a lot more, and then I didn't react much to little things like a happy moment and rarely a sad moment in a movie/show. Now I struggle to show any emotion at all, I just feel sadness and anger like you described, that "burning" sensation. The last time I outright cried was when I was very sick with a respiratory infection and was in a lot of pain. It eventually forced me to cry because I couldn't handle the physical pain. It hurt so much to cry then, but a couple hours after I felt better.

I know I purposely learned to stifle my emotions because I was showing them all the time and it was very overwhelming not just for me but for anyone close to me. I have very vague memories of situations where I could feel emotions starting to get strong to where I'd show them and purposefully telling myself "no, you will not show it". Now it's just natural for me to bottle them up. My theory now is that I've filled the bottle and emotions leek out when they're being invoked through something I'm watching.

I can't really offer any advice unfortunately as I've not found a way to process the emotions properly, just here to tell you that you're not alone in the way you feel.