In my own attempts to understand my emotions better and give them proper attention, I've found that it's helpful to:

1) explicitly state what emotions I'm feeling (this is surprisingly difficult for me)

2) figure out why I'm feeling that way.

Logic and rigorous thinking becomes relevant for that second step - it's where the "both / and" union of the two modes happens.

This process forms the backbone of "Nonviolent Communication": https://www.nonviolentcommunication.com/

If you want to have effective, kind conflicts, that book is a must-read.