Nitin, yes, you can definitely have bookmarks (and all the other types of posts) be ‘published’ but not visible to others. WordPress comes with several options, published (public), publish (password protected), and published (private). It’s the password protected one that shows up visibly on your site and asks for a password that you considered ugly. Thus you can publish a post privately and it will only show up to you (and other admins of the site) while you’re logged in; the general public won’t see it and there won’t be a password option either. These are all reasonably well documented here: https://codex.wordpress.org/Content_Visibility
Beyond this there are two other options for places you can ‘hide’ these posts without them being published: either as ‘drafts’ or as ‘pending’ posts. These options however don’t necessarily (easily) preserve the publication date for when you posted them, but they are options. I suspect someone’s got a plugin that may provide additional options, but I’m not specifically aware of one.
Personally, I use all of the above options in a fairly on-label use. I probably have far more private posts than public and often use the meta-data (tags, categories, etc.) on them for searching/reading.