Setting out to do so is the first and hardest step to take, so congrats! But, of course, the work doesn't end here. As I understand it, someone who's reasonable means one who can be reasoned with, i.e. someone who accepts and occasionally yields to persuasion attempts, and doesn't shut others off through obstinacy or abrasive, uncooperative treatment. In some ways it's the antonym of intransigence. It can also mean someone who possesses enough common sense to facilitate interactions based on a shared view of how the world works.

You may reduce your likelihood of showing such tendencies if you reframe social interactions that involve arguing in a way such that being (acknowledged as) right is less important than maintaining harmony. What some people, the kind who drag out arguments in the name of truth or rightness, don't understand about arguing is that the interaction of arguing takes place in a social context, is awarded limited time and patience (!) before it starts getting on people's nerves (so no, it cannot be prolonged indefinitely until truth finally prevails, however long that may take), and may not be worth the hostility most of the time. Developing some more empathy and thinking about what the other person seeks in the interaction, and whether you're giving it to them, may be of help.

There's a kind of trick that may be of help, but it has to be culturally shared for it to work. You know how LessWrong has some local proverbs such as the Litany of Tarski or Tsuyoku Naritai that people can invoke, but only to other LessWrongers, to remind them of shared values that should prompt an improvement in their behaviour? It would be nice if there were some appeal to being understanding or reasonable that carried the same tone of solemnity. Something that essentially means "I know I can get biased and unreasonable occasionally, but I am committed to the values underpinning collective truth-seeking, and I pledge to allow others to remind me of my commitment, and to attempt to yield when they do so". But in a pithier form.