I'm a fellow introvert, so I feel your pain. It sounds like you've got a great handle on your strengths and weaknesses. I know what it's like to work in a less than ideal environment. Your current bosses sound like jerks. There's being extroverted, and then being a bull in a china shop. I work with a couple of those and have learned to not put up with their BS. I've found that people like that tend to view introversion as a weakness, when it's not. They prey on that weakness and try to exploit it. Don't let them. You are strong and good at what you do.
I'm a bit of a contradiction when it comes to introversion. I am in no way an extrovert, but can act like one if I need to. People tend to look at me as a leader, and I've taken that role many times, including serving as president of two different professional board of directors for several years. When I say I can act like an extrovert, I'm not talking about the obnoxious, arrogant, effusive type, but rather a more quiet type who knows how to call a meeting to order and tell people to quiet down when needed. I'm a good delegator and organizer and know how to make matter-of-fact statements with confidence, even if I don't feel it. This was not always the case. I've worked in plenty of places where I was railroaded for being quiet and not part of the 'in' crowd. I didn't know how to speak up and stand up for myself back then, but I've since learned how, even though it still causes me a lot of anxiety.
Cassie's got some great advice as far as using scripts or email and rehearsing situations prior to encountering them. Sometimes I get very thrown when something goes off script. Asking open-ended questions designed to get people to talk about themselves is a great idea too. As Cassie said, people love to talk about themselves.
I totally get what you mean about being more extroverted with your friends. I'll chat for hours with good friends of mine and people I feel comfortable with. With strangers... not so much.
I can't help you with your first question. All I can say is apply to as many jobs as possible and don't be afraid to negotiate and clearly state your needs. As far as your second question... I'll tell you what I tell my students. Fake it. You may walk into a meeting feeling like you don't know anything and it's going to be like walking a gauntlet, but keep your head up and speak with confidence and authority, even if you don't feel it inside. I know... easier said than done. And I know how exhausting it is, too. But you may need to do that until you find your ideal job where you won't have to.
Best of luck. I hope this helps.