How do I get more engagement with my content on LinkedIn?

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.

By content, do you mean things like updates/posts and stuff you share, or written content like Pulse posts?

Cold messaging on LinkedIn isn't going to drive engagement. You'll likely just be annoying the person receiving that message. Speaking as someone who's on LinkedIn as a business owner, I get tons of cold messages from people trying to sell me things, find a job, or ask if I can "just take a look" at their resume or project without paying for my time if I don't get some lame form e-mail I just throw in the trash. Not going to work!

You need content that speaks to your target audience. Are you creating and sharing things they'll find interesting and helpful? Something that hasn't been seen before?

Find out what your LinkedIn audience wants and deliver it to them whether it's via update post or Pulse. Once you get some engagement, try messaging people who've interacted with you by pointing them to your blog and other off-site stuff through direct message.

1. Engage people by publishing UNIQUE, provoking, controversial content on linked-in. Don't post the same stuff that everyone else does, like "10 traits of leadership".

2. Don't cold message someone. Look at the connections of your connections. If some look like they would be valuable, then ask for an introduction.

Identify the topics and subject that you know well enough so that when you write (share updates), the community trusts that you know the subject. Building 'trust' is the first step towards engagement, and it is as true for individuals as it is for organizations.

Once you share an update, add your own perspective for the right CTA. Why should anyone engage with the post - are you giving them some food for thought? Are you questioning some industry recommended practice? Are you seeking opinion?

When you combine (a) your expertise that builds trust (b) Right CTA, people will be encouraged to contribute.

Important: Tag the sources as required. It adds authenticity to what you post. And then be timely and courteous while responding via comments.

Sure, you can get more views on your LinkedIn Content if you take care of following things -
1) Write an eye-catchy tagline for your content
2) Keep your content short and meaningful
3) Try to help others through your content with some useful information.
4) Share your content on your social media.
5) Try to add a recent fact/news about that particular topic.
6) Add relevant image/video.
7) Add a link to your landing page or website or blog for users to get more information using a Call-To-Action Button.