We all communicate differently, and some of us are better at it than others. However, at the heart of communicating is a genuine willingness to examine issues and problems in a non-judgemental way - if we enter into a conversation with changing the other person being our goal, we will meet with a lot of resistance. When we are used to exhibiting all our emotions during conflict or important discussions, this seems to be almost impossible, but there are some techniques that might be useful during these times

Think of a scenario which may result in conflict - I’ll use an example which is relatively minor (but this technique can be used in conflicts of different sizes): You’ve asked your partner to pick up some milk and bread on the way home. And they’ve forgotten.

Try observing what has happened, rather than add your interpretation, e.g. “You forgot the milk and bread.” - vs - “I ask you to do one simple thing, and you forgot. You don’t care about feeding this family.” Factual statements, without assumption and evaluation.
What you feel needs to go beyond your thoughts. Unexpressed emotions can and will spill out in other ways. Understand how and why you are feeling the way you do about the situation, and let your partner know in a non-judgemental way. In this case you could say “I’m frustrated as I needed those things and didn’t want to make an extra trip to the store if you were going to be passing by.” Own your own emotional response.
You need to be able to understand what it is you need and be able to express it. If you do this, you also give your partner room to respond - are they able to meet those needs, and do they want to? For example, you could say “When you forget it feels like you aren’t treating me with consideration.”
You’ve talked about facts, feelings and needs. The next step is to make a clear, unequivocal request - what can your partner do to make sure you feel your needs are met? Maybe you could say: “I’d like you to tell me if it isn’t possible to stop at the store when I ask you to do it. If you say yes, I’d like you to set yourself a note to remember.”

This process takes some getting used to - it can feel strange setting emotional reactions aside, but remember - you ARE communicating your emotions during these stages, but taking the fire out of what you say so that how you feel and what you need are crystal clear.

The more you do it, the more natural it will feel - neither of you are perfect, and communicating in a de-escalated way will show acceptance of each other’s mistakes while still asking for what you need.

Remove the heat out of communication, not the heart.