Hi!
Firstly, congratulations on your dissertation result - that's excellent and you should be very proud! It is a little unusual to publish undergraduate work but your tutors wouldn't be suggesting it if they didn't think your dissertation had the potential - so this is fantastic news and you should certainly consider the opportunity!
I'm in a similar position to you. I'm an English student and am currently in the first year of my PhD and hoping to move into academia on completion. One of my MA tutors suggested that an essay of mine had the potential for publication and suggested a couple of suitable journals that I might like to target. Once I'd settled into the PhD, I decided to look into this with a view to making the essay into my first submission to a journal. Going back to look at the MA essay again with 8 months of further study under my belt, I'm still really proud of it - and I agree that it does have the potential to become an article. But having read more journal articles myself, I can see areas where it lacks the necessary depth of analysis, or where the tone isn't quite right. Ultimately I've had to do two substantial re-writes (I'm currently on my third set of edits for it) to move it from being a strong academic essay to what will, hopefully, become a good journal article.
My PhD supervisors and the tutor who originally suggested I expand the piece for publication have all kindly provided comments to assist me in my re-writes and edits - these have been invaluable so I would definitely suggest asking your dissertation supervisor if they would be happy to comment further on the piece as you develop it, and possibly get another tutor (someone on your MA maybe?) on board as well. It might also help to ask them if they have a particular journal in mind that they think your dissertation would be a good fit for, and what they feel you would need to do to turn your dissertation into an article.
I would then suggest reading some articles from the journals suggested (or any other journals you think you might want to target) to get a sense of what their articles include, how the arguments in them tend to be laid out, house style, and the balance of source analysis to general commentary that the journal prefers. They'll probably also have submission guidelines on their website that will tell you how they want their references set out, what word count articles need to be etc.
My final piece of advice would be not to rush to publication. It's tempting - especially during your MA year - to race to publish something that will help you in making PhD applications. One of the best pieces of advice I received was that it's better to publish a polished article in a good, well-recognised journal within your field than to race to get into the first journal you can find. There's an element of quality as opposed to just quantity when it comes to building a CV of publications.
So, by all means, look at turning your dissertation into an article - get the advice, work with your tutors and be prepared to do the re-writes - but don't fret if you don't end up publishing it in the immediate future. Concentrating on making it the best first publication that it can be will, ultimately, pay dividends in the long run. In the meantime, you can always begin to build up some publication credits during your MA year by contributing academic book reviews or conference reports, whilst working on polishing your article.
Hope that helps, best of luck with the writing, congratulations again on your dissertation result, and enjoy the MA!
Amy