I work in procurement. That means I negotiate with suppliers all day, every day. I'm the one the team calls when they need something fixed, and want something better than using a big hammer or lighting everything on fire.
The absolute best tip was courtesy of my first boss in the field:
"Always be prepared to walk away and say 'No, thank you'. "
You won't look weak (except to the wrong people). Instead you'll come across as empowered, that you know what you want, have no tolerance for games, but also can be trusted. Very few people like people who always say yes, no matter what anxiety says. In order to be here, it's important to have options and choices you can accept. Be ready to jump rails if it looks like you won't get your first choice. That's how good negotiators in my field do it - they have several acceptable options.
It's still not easy to do. It requires perseverance and flexibility. Allow yourself to practice in lower-risk situations.
Can confirm. My local group assumes you are queer and neurospicy until you state otherwise.