If you think they're primarily interested in finding ways to mischaracterize you, you're probably mischaracterizing them. Intentional straw men happen much less often than accusations thereof. When they do happen, I find it's best to draw them out and expose their actions as being an ass. More likely you haven't been clear enough and they haven't gotten exactly what your argument is, but they think they have and are responding to the wrong one honestly. In which case, storming off makes you look like an ass.
Because different belief systems are often at odds with each other. Often over different approaches to policy, but occasionally one belief system thinks another is going to be tortured forever in the fiery cantos of perdition. Puts a damper on genteel debate, that bit does.
Until both parties understand how the other thinks, they'll never be able to drill down to the heart of the matter nor find a suitable compromise. They won't even be able to begin to do so, because they'll at best be knocking down unintentional strawmen while talking past each other, each perfectly convinced of their own correctness.
Plus, claiming to end the conversation when you understand someone makes a conveniently perfect excuse for ending a conversation long before you understand someone, but when it's just not going they way you'd like it to.
There's probably an actual name for it, but I call that the "just us chickens" argument. Same thing happens when YHVH, Creator, omniscient, omnipotent when only the faithful are in the room, becomes the ineffable, inscrutable, undetectable Intelligent Designer as soon as someone has the bad manners to bring up science.