Hi Donna, great question!
It sounds like you've already started with evaluating your audience. From your previous post I see that you are a cookie company so generating demand for your yummy cookies could just be as simple as spreading the word (and the frosting). Knowing the LinkedIn is mostly a business audience, I'm assuming you're looking for corporate catering or recognition types of customers.
You'll be able to share your emails onto LinkedIn as well as your other social channels. This article has some best practices to keep in mind when creating your post. I'd add that you'll probably want to custom your social post for LinkedIn to start the thought for the reader. "Let us handle dessert for your next company lunch", "Nothing says employee appreciation like a cookie" - something that, in front of the right person, will help make the memory that cookies are a great idea! I'd say you could also be ahead of the curve around holiday planning. Employee appreciation day, customer service day, national boss's day. There are lots of little holidays that you could start sharing about early to help people be prepared on the day itself.
If the organic post route isn't cutting it you can also explore running ads on LinkedIn. It wouldn't necessarily be through Constant Contact, but for the right businesses and situations the ads can be valuable.
We'd love it if you could share what you try, what you found works, or what doesn't - and I'd love for anyone else with lots of experience around LinkedIn could share as well!