In order to be a database administrator, you must possess a wide variety of skills. You appear to have a good variety of skills now.
Instead of trying to become a "junior DBA" and marketing yourself as such, consider becoming an "IT Generalist/gopher with a keen interest in databases". If you make it clear that you're looking for database experience, most employers are willing to work with you.
The database is where the "crown jewels" of a business are stored, so it tends to be jealously guarded and is rarely entrusted to a "junior" anything. There is just too much "high value" data concentrated in one place to let any manager be comfortable with anything less than the best support that they can get. Many businesses have horror stories about "the old days" when somone foolishly deleted, damaged, destroyed part or all of the database and crippled the company for days.
What you are doing is a great idea, the experience will be very valuable as you get started with your career, but you need to have some patience. You need to think from the perspective of the business and understand why they are wary. The search for this job means that you'll need to examine yourself and your own motives and find a way to make both palatable to the management team of a business, and that may teach you as many useful skills as the last year or two of school (and it might take equivalent effort)!
In theory, theory and practice are identical. In practice, theory and practice are unrelated.