Don't try to point out why they are wrong, but rather why you believe what you believe. And when they tell you why they believe what they do, no matter how ridiculous it might seem, respect their opinion and explain which points exactly you disagree with and why.

I think the idea of trying to convince the other is flawed in itself. It implies that you are right and they are wrong. Approach any conversation with that mindset, and neither their nor your opinion will change.

Instead, try to see a discussion as a way to exchange perspectives with the goal of finding the truth. Only if you are open to the idea of changing your mind can you hope to change that of your conversation partner.

I remember getting into political arguments that went nowhere at the time but resulted in me changing my mind years later. The people I argued with never knew about my change of heart. As far as they knew I was one of those people who get more entrenched in their beliefs.

What I'm getting at is that your arguments can hit home without looking like it. What you're seeing as getting defensive could just be the early stages for them changing their minds.

This can be especially true if someone's political beliefs are part of their identity. You don't make those kind of changes all at once.

So I'd say just argue in good faith, don't try to score points, provide food for thought if you can, and hope for the other person to eventually find their way to the truth.

Have you heard of reactance Theory?

It basically states that people will react to a something they perceive as a challenge of something they identify with, or perceive as a loss of freedom, with reactivity and defensiveness.

Rather than explaining your point of view, you approach and ask their opinions with curiosity. Listen openly.

Many people can't imagine different perspective due to their limited intuition, they need concrete examples of improvements so they can imitate these. In other words, if your intention is to convince someone about your improvement, show to that person how the improvement works well on you, without impose your vision directly.

You can't the mind that refuses to change.

"Against stupidity, the gods themselves, contend in vain." Friedrich Schiller

Or, 'You can lead a horse to water, but you can't make him drink.'

Don't waste time with the people who refuse to listen; work on building a network of like minded people.

What can someone who lives in an apartment building do? Or a student in a dorm? Which candidates should we support.

Your time is a limited resource, don't waste it pounding deaf ears.

As much as I hate to say it, I think learning to make use of the same communication strategies that bad-faith argument people tend to use is the way to go. In specific:

1. You're not trying to "win" against the person you're talking to. That is, in a lot of cases, impossible. You're trying to communicate your message to an audience (which, sure, includes the person you're talking to). You're not limited to the framework the person is trying to give you. You can just say your message.
2. Short, tangible talking points can be more effective than detailed and accurate explanations. Pick one thing that's true that encapsulates a little example that's hard to deny about the way you're saying the world is, and stick to it. This technique is so effective that it can create a narrative even if the overall world-picture it creates isn't actually accurate; if the narrative you're trying to construct actually matches with reality, then you'll have a broader base of little exemplar anecdotes to draw from.
3. Stay patient, don't get irritated, don't feel like you have to "respond" to everything. If the other person's talking in good faith and asks a relevant question, then sure, you should answer it. But if they're just doing a Gish Gallop or something, it may be more effective to call out the Gish Gallop and keep explaining your own POV than to try to debunk every single thing, or to get them to "agree" or pin them down to something they're saying that's wrong, when they're not interested in cooperating and the effort will just derail you from making your own side of the argument.

Showing instead of convincing. Everybody can talk, what counts is the tangible outcome.

It goes hand in hand with avoiding putting someone in the defensive. I find it hard sometimes, I usually make my opinion clear but try to change topic and remain open to talk about other stuff. That's not great and I'd like to be more skillful in being silent.

https://anticar.org/

Trove of resources.

In terms of dialogue, you can't beat Street Epistemology