These are two different questions. I will try answering them one by one.
1) How to get more content engagement?
Content can be your LinkedIn updates, contributions to groups, or articles written on LinkedIn. Since LinkedIn is a professional network, reach and engagement (except in case of Groups) is limited to your network. So, if you want to improve engagement, you have to first improve your connection count. Once you have done that, you have to focus on raising your content game. On LinkedIn, people value insights served in fancy words.

2) What's the best way to cold message someone without it sounding random?
Cold messages always seem out of the place. So, why not build a connection before dropping a message. Once it has been made, you can thank the person and give just a whiff of what you have in mind. The person will get the hint and inquire if he (or she) is interested. If your request is not being accepted, you can upgrade to a premium Membership and craft a killer message to quickly convey what you have in mind. The message is the tricky part but if done right, it will get the job done.