Nastia, you mentioned having a secure relationship with a partner, who is presumably with you at the burn. That person is probably aware that you tend towards shyness but actively want to break out and interact and make connections with people. I recommend that you make it an early-week goal for the two of you to enter into situations together where you'll meet people (volunteering, helping neighbors set up, and offering gifts as others have suggested are all great ways to do so). Have you partner there for backup and encouragement, but don't depend on them as a crutch. Look at it as practice, becoming more comfortable out of your shell. Once you're more in the swing of the Burn, make it a goal to strike out on your own--maybe with some of the people you've met, or maybe into totally unknown social territory. Make your intentions clear to your partner before the Burn, so that they can support you in your goals and also won't feel like you're trying to ditch them once you're feeling more confident.
If you feel like the problem is that others are intimidated by you, focus on having fun. Dance, laugh, play. If you're smiling and joyful, other people will not be intimidated.
One thing that I really appreciate on playa are brief but emotional connections. These mostly happen to me at the Temple. Everyone has their own opinions about what the Temple is and should be, but for me it holds heavy significance. I'm not a spiritual, woo-woo, or overly open/emotional person, but it's the one place that breaks me down. I've placed tributes to lost loved ones there every year, and I mourn more deeply there than anywhere else. To be honest, I bawl uncontrollably (though thankfully it's usually quiet) in the Temple at least once a year. In that space, I have always found comfort from others, who are happy to give it. It usually starts with eye contact from across the crowd, often with someone else who has experienced loss or is feeling overwhelmed. One of us will approach the other, we'll hug, cry, sit for a while, and then talk. I open up in a lot of ways that I struggle to in default. I think that the Temple opens a lot of people up by making them feel exposed and vulnerable but at the same time that they are in a safe place, surrounded by good people who care and genuinely want to connect too. Even if I'm not sad, if I ever feel isolated or lonely at the Burn (which can definitely happen, regardless of how many people are around), that's where I go.