I'm a copywriter but quite an old one and have never done a course in my life, but there are advertising/PR courses you can go on (normally degree level and length though!)
What sort of writing do you have to do now? Is it sales/promotional - ie designed to elicit a response in the reader to buy a product/visit a website/ask for more information? Is it aimed at other companies (business to business, or b2b) or at the end user (business to consumer, or b2c)? Or is it technical copy in the sense that you need to be skilled or practised in what you're writing about, not just in writing itself? Are you good at thinking up clever headlines and/or persuasive body copy? (Not everyone is equally at ease with both).
What you'll need to get an interest from a potential employer is a portfolio of your work (this can be digital/pdfs etc - not necessarily an actual portfolio) so they can see what your style is. If you don't have any examples then start writing some!
As well as being a copywriter I have hired freelancers/juniors and what I always look for is
a) someone who can spell (you'd be amazed at how many copywriters there are who can't)
b) someone who can put themselves into the head of the reader/viewer/listener
c) someone who can quickly grasp the most important point that needs communicating - the USP that will persuade the audience that they need look no further than X brand because it offers exactly what they're after.
d) Someone who can say it in fewer, rather than more words (not that you'd believe me looking at this post)and get to the point as quickly as possible.
Someone else might be able to advise you on courses but I would just say, find a style you admire and want to emulate and work on producing something that gives you the same feeling of satisfaction.