You sound like an incredibly beautiful sensitive soul and I imagine that seeing how easy your grandmother made it for you to love her, she too must have been this way.
I know this is easier said than done but you can't blame yourself for the choices your grandmother made or the choices her doctors made before she passed. If you were to ask her, in your imagination 'What could I have done to change your mind if I knew that pushing you to see a doctor earlier would have helped?', can you imagine her smiling saying to you 'There is not much you could have said to me'?
If you were to find a quiet space on your own, can you imagine having certain conversations with your grandmother? They could be simple positive conversations, such as 'Which top will I wear today?' or 'What positive difference can I make to my Mum/Dad (her child) regarding having lost you in their life?' Imagine what she would say to you in response. I know this may sound strange but it's how some live with the person they have physically lost still in their life. Maybe you already do this to some degree, talk to her.
H-c, I regard myself as a sensitive person in a lot of different ways. While being sensitive comes with its challenges, there are also benefits. The heart sensation you experience is an interesting one. Personally, I find that when someone I care about has deeply upset me I can feel what's like a heaviness in my chest (the only way I can think to describe this feeling). The phrase 'To feel heavy hearted' is most appropriate. Then I can think of someone who leads me to smile and suddenly there is more of a light sensation (light heartedness). For you, it sounds like you are experiencing heartache. You actually feel it. This is why I suspect you are an incredibly sensitive person. Consider opening your heart and mind to the conversations I mentioned you trying with your grandmother and see if this makes any difference. It's kind of like a form of meditation.
Interesting, all those sayings that are out there. Once you start to focus on your sensitivity to certain situations or people a lot of those sayings become relateable. Another one 'To get choked up'. Whether we're feeling this sensation in our throat due to grief and the need to vent (through tears or counseling) or we feel it when someone is challenging us to find a level of healthy self-esteem due to their abuse (venting through communicating what we will no longer tolerate), it helps to identify our feelings.