I think you are wrong about how off-putting your tears are.

In my experience, when someone loses someone they care about, it is comforting for them for other people to be sad and feel the loss as well. I think that if instead of focusing on your own tears and your embarrassment when you get emotional, if you still focused on the other person while you were crying, they wouldn’t feel obligated to comfort you. Then they could just continue to share with you and be comforted by the fact that you empathized and were moved.

When my brother and SiL had a stillborn baby, I went to visit them. They genuinely seemed somewhat relieved to see me crying while we visited together.