That is one of the hardest things to do but it's what it takes to get past a fight. In order to get what you need, or to have a chance at it, you need to be able to hold your spouse in compassion with yourself, while at the same time telling your spouse what your hurts are and asking for what you need.
There are so many places where that can go wrong.
When we hurt we usually tell someone else what they are doing, what they care about, what they are thinking, or worse - where they are wrong and where we are right. I often see those as authentic attempts to 'say what we feel' and ask for what we want instead, though muddied by defense and fear, and we end up criticizing, attacking, maligning, and judging instead - and all the while sounding angry, bitter, resentful or cold.
Focus on feeling, not facts.
Tell your spouse what you feel, not what they did. Feelings are not refutable but narratives are. People spend hours and days fighting in what I call courtroom language, arguing over whose version of a story is right, assuming that rightness is the goal. Rightness is just another form of righteousness and righteousness is often not tolerant of mistakes but love and forgiveness are. When you can say "I feel hurt" instead of "I was right" you're on a better foundation for loving reconciliation.
Be aware of how you might look when you're upset.
When you're hurt you won't register what you, yourself, look like. How you look and sound to your spouse will affect how your spouse hears you or sees you. Are you able to describe what an angry-you looks like to someone else? Try that when you're in a good mood so you're less defensive. How does your spouse see you when you're hurt? Will they see someone needing their help or concern or will they see someone going for their throat? In other words will they be able to hear your hurt or simply fear their own hurt?
See where the other is coming from.
Everyone has their weak spot(s). That weak spot is the concern you fear most, the slight you react the most to, the shame you avoid at all costs. What is your spouse's? Chances are, when you are mad, they are reacting from this weak spot when they see an angry-you. Chances are you are protecting or soothing your weak-spot when you become an angry-you. Can you see your hurt weak-spot and your spouse's at the same time? Everyone generally has the ability, but not when while angry.
Tell the other what you need.
Behind the short sighted reactivity of an angry spouse, and the short-sighted reactivity of an angry-you are those the weak spots that need kindness, love, attention, and support as well as the parts of you and your spouse that do the loving. Try to speak to the loving part of your spouse and tell their I-love-you part what you need. It's in there. And when it hears you, coming from both a place of love and need (not anger and conviction), it will hear and hopefully your spouse will soften. Some shells are harder than others, so some spouses will take longer to soften.
You're doing pretty well when you can hold your own hurt enough, that you can express a little compassion for how your spouse experiences you, and then say what your feel and still ask for what you need moving forward. That's pretty hard to do. I'm happy to help you get there.