How do you stop getting defensive and going cold when your partner disagrees with you?

I’ve noticed a pattern in myself that I really want to work on.
When my partner and I have different opinions, especially in person, I get defensive very quickly. I argue my point repeatedly because I feel the need to explain myself or make him understand where I’m coming from.

If I’m tired of the conversation, I do the opposite. I shut down, go quiet and become cold.
The problem is that even when I recognise what’s happening and say something like, “Okay, we don’t agree, let’s leave it here,” emotionally I can’t seem to actually leave it there.
I’ll still feel annoyed afterwards. I’ll become quiet/withdraw my affection, the vibes becomes awkward, and I’ll still have this strong urge to bring the topic up again because I thought of something else I want to say or because it doesn’t feel “resolved” to me.

I don’t want my partner to feel like he has to agree with me or walk on eggshells around me. I also don’t want to suppress my opinions just to avoid conflict.

For anyone who used to react similarly, how did you learn to catch the defensiveness before reacting? And how did you learn to tolerate the unresolved/annoyed feeling afterwards without either reopening the argument or going cold?

I'd appreciate any advice, thank you.