I'm a verified twitter user. I hope my advice helps!
What do you mean when you say you engage? There are many ways to do this, and how you use it will influence how they see your activity on the platform.
You should reach out to people in your niche and retweet and contact them, and do so in a way that is almost never about promoting your brand (that would be spammy).
But even if you are the best twitter user around, ultimately the purpose of verification is because you have some form of celebrity, where people imitating you is a genuine concern. So if you have come across accounts that claim to be you, or websites that use your face (a problem I was facing), then that is important to use, as the whole point is that the tick mark shows that you are the real you compared to any potential fakers.
For the question of losing followers, that's troubling, and even if you never paid for them if they are leaving in those amounts it may also be a case where your account is randomly followed by fake accounts to make them seem real and other people will pay them to get a follow (so they don't exist to follow one accont, which would be more suspicious). If twitter sees that a lot of people are unfollowing you, even if it's not your fault, then that will hurt your chances. You may want to contact their support separately to their verification department and see what's going on there and solve that problem first and foremost.
If you have further questions about maintaining authority online, which will help the verification process immensely, feel free to give me a call!