Your question reminds me of the good old 'must have experience' job ads for uni students. How people can gain experience without a chance is beyond me. Similarly, unpublished authors face the giant hurdle of credibility and exposure that can feel extremely isolating, so I do have some tips to help you through this.
I think the first thing to do is to stop pigeonholing yourself as an 'unpublished' author. If you are writing books for children, you are already an author, and it frustrates me when writers feel the need to delineate between being 'published' or 'unpublished'. And never refer to yourself as 'desperate' even when it's tongue-in-cheek. Sit calmly and centered in the knowledge that you have what it takes, that you will be noticed eventually, and simply send that intention out there--in thought and action.
My second tip is to have patience. Becoming published is (usually) a very long and convoluted process. There are thousands upon thousands of emerging authors and only a small handful of books are published each year (by comparison). For example, a large publisher might receive 10,000 unsolicited manuscripts per year but perhaps only twenty (or less) of those go on to become books.
The really important thing to remember is that receiving rejections is not necessarily a marker of your talent. Published and even very successful authors still receive rejections--and the reasons are oftentimes nothing to do with the manuscript's publishability, but rather its fit with the publisher, their current list, if there's space on the list, if they have nothing else like it on their list, if it fits with their current market direction, if they feel it will sell, if it's cost-efficient, etc, etc. Again, stay secure in your self-belief that your manuscripts will 'fit' somewhere, sometime. You just have to keep searching for that fit.
This means you should submit as often as possible. Keep your communications warm, professional, succinct and not overly self-promotional. Be sure to research your target publisher really well, and just keep trying. The more publishers see your name, the more you will 'stick' in their mind.
Creating a strong online presence is absolutely vital, yes, but not in that 'stand on a chair and holler' kind of way. Be consistently present but stay humble, professional and true to yourself. Also be sure to align that presence with your 'brand' and with the type of work you create. If you do this, very soon, people will automatically associate your name with the type of work you do ... if you hear the name Andy Griffiths, what kind of books do you picture? This is author branding.
On your blog or website, talk about your works in process (WIPs). Give sneak peeks of your work and talk about your writing processes--essentially, share what you're doing and give something to your audience. Maybe even talk about your journey--how you're feeling, how vulnerable you feel, what you're submitting, etc. Show you are actively writing and are serious about creating great books. I hate that term 'fake it till you make it' but, in essence, behave as though you are already an author ... because you are.
I would highly recommend social media sites and using them for solid networking, but in a way that allows you to get to know people and form genuine relationships (some authors make the mistake of using them as nothing more than a megaphone platform for their work). If you form relationships, you are much more likely to be noticed, to be talked about positively, and to take advantage of connections, opportunities and even collaborations that could lead to contracts.
Visit schools and daycare centres and community events to read books or even your own work. You could also talk to adults about being an author and your publishing journey.
I also recommend attending book launches, festivals, conferences, and anywhere you can meet industry folk--getting to know them and chatting with them directly (without stalking!). Use your intuition with people and take things slow. Publishers love authors who are easy to work with and don't come across as demanding or complicated.
Apply for emerging author grants. Enter competitions. Publishers often judge these, and this is also a great way to have your work seen. You can also have your work critiqued by publishers at conferences and festivals. This will also get your name out there.
I know that feeling of wanting to be published so badly, it hurts. I still feel it with my new works, and I understand how overwhelming it is. To get my name out there, I would definitely focus on an online presence and networking (and building relationships), backed up with lots of subbing. Getting involved in the industry by supporting others and 'giving back' (as I have done with Kids' Book Review and the 52-Week Illustration Challenge) also gets your name out there.
I have read your work, Michelle, and you really have what it takes. I hope this helps on your publishing journey--it's only a matter of time!