Cold messages to people in my field, do they work? - any advice?

I sent 12 messages on a professional network and got two replies. There are 12 people counting on me, so I am trying to be careful. I worry about bothering people.

Follow up with a thank-you and a short update. One of my chats led to a referral a month later.

Do not apologise for asking. Most people remember someone who helped them.