Very true. In fact, I don't think people even start out with "beliefs". They start out with feelings. The emotional response is the equivalent to an axiom, and is abstract and formless. They must use their life experiences to form a reasoning to justify why they feel a certain way, and that is their "belief". This justification is rarely entirely accurate, it's just an approximation filtered through their own biases and reasoning capabilities.

When I have an argument, I find it most productive to first seek out to understand how the other person feels. Peoples' feelings are rarely "incorrect", and usually align with my own feelings at some fundamental level, although perhaps with different intensities. I then try to work with them to put our brains and experiences together to find a more solid understanding for why we feel that way.

What is a belief but an axiom? A prior that must be assumed, as it cannot be proven within the system.

I will definitely concede that the logic people use on a day-to-day basis is much fuzzier than an analytic might prefer, and that there's much greater allowance for contradiction (see the phenomenon of Cognitive Dissonance). But I do think that most people argue from their own first principles, consciously or not. There are rare cases where you can convince someone of something when you discover and correct a logical flaw they have made. More often than not, though, a 'logical argument' is a veiled battle of dogma; both participants are moderately rational, and share the same rules of logic, but their axioms are so alien to eachother that what is plain as day to one is a violation of human decency to another. I'd reckon that in reality, it's rare that two people are actually even arguing about the same thing, even though they may be using some of the same words.

Agreed, I don't think people really debate emotionally charged issues from first principles. I think it's an error to think of this as a matter of logical axioms. That's not how people (usually) debate.

Its obvious that logic isn't a good model for how people form and change opinions. (I just feel that it isn't)

So... what's the state of the art?

It's been out of favor since Derrida, but I think Structuralism is due for another pass in the limelight. An extreme subjectivism about reality, shaped by the stimuli you are exposed to. These external stimuli are what define your axioms, and from there, you use the same (fuzzy, bad, and prone to inducing contradctions) logic as everyone else. It provides a compelling model for why people have difficulty reconciling their opinions; they're not even talking about the same things, just using the same words, the same signs.