Treat people with dignity, and they will likely do the same to you.
When you're speaking to people, try and talk a bit slower and more calmly than you would. Try to control your tone and volume, and try to make your points clear. When you are talking about something that you're not 100% sure on, make that known. Say, "I'm not sure on that, I'll have to check on it.", or "I honestly don't know enough about it, but I can check".
Many of the conversations I had at my previous jobs were done digitally via text. Therefore, it is incredibly important to understand how people read text conversations. If you're typing out several paragraphs, it's especially worth it to read through it before hitting send. Many times have I initially written emails with sarcastic quips, only to realize how that could sound once sent. Leave out personal details - if something is important enough to you, speak with that person on a phone call, or go see them in person.
It's good to go into a conversation with an open mind and proceed by ignoring sarcasm and false niceties, but if someone is seeking drama, don't allow it. Remove yourself from the conversation. "Can we pick this up another time? I need to go do X thing." is a perfectly reasonable thing to say.
Many senior programmers I've met that are stuck in their ways usually think of ideas as attacks on their code. I have received a lot of responses from people like this that are one-liners meant to discourage further messages. "No, we don't do that here." and "That won't/doesn't work." are examples of being exact and making a strict assumption. Stop thinking this way. If there is anything you take from this post, take this: Stop being exact, and stop making assumptions. You have a chance of being wrong. Hardware and software changes all the time.
To add to the above, make sure you ask good questions when you're talking to someone. Sometimes, marketing (or another department) may actually not know exactly what they want. You can help them by actually being a part of the conversation and by asking meaningful questions.