I can't say what you "should" do, but here are some thoughts.
Your name is really really hard for English-only speakers. Sorry.
I have a friend named Nhan. I put a lot of work into learning to say his name correctly. Most people did not. After two degrees in Canada, when he moved to the Valley he gave up and changed to an English name. An ex-co-worker named Nhi has just accepted that everyone calls her "Ni" (like "knee")... at least it's closer than calling you "naan" or "non". ;P As far as I can tell, the best spelling to get people to say your name correctly is probably "nyen". (Like maybe if you put in your email sig "Nhan (pronounced like 'Nyen')") If you want to make a joke about it, I guess you could even say "the 'h' looks like an upside-down 'y', so just pretend it's a 'y'". Although I dunno if you feel okay about using humour about your native language to try to improve communication.
As I'm sure you've noticed, Vietnamese pronunciation rules have surprisingly little in common with English pronunciation rules, so virtually zero English speakers will correctly pronounce your name, unless they've been told (by you or by someone else). The other more-common examples are Ng and Nguyen (I've been told these are actually Chinese names, but everyone I've met with one of these names was Vietnamese... I dunno, I'm just an ignorant Canadian struggling to be less ignorant). Almost zero English speakers pronounce these names correctly, even after being told many times! It's not that hard, people! But most people are hilariously lazy about learning new sounds. "Nhan" is a lot easier to get almost-right than "Ng" or "Nguyen".
Personally, as a language nerd, I like it when we're forced to learn a little tiny bit about another language, so selfishly I hope you keep your name. =] But you'll have to decide what's best for you.
As a little side story, in Vancouver there used to be a pho restaurant named Pho Bich Nga, on a very busy street (Kingsway, for the Vancouverites in the crowd, which has lots of Vietnamese shops... I'll drop a shout-out here for Green Lemongrass). I'm told that's correctly pronounced kind of like "fuh beek nga". English speakers don't really know what to do with that, but the best guess usually ends up being "fo' bitch nigga". Teenagers think that name is pretty funny, yes sir. Sadly, they changed the name, and a Vancouver cultural landmark is no more.
https://www.google.com/search?q=pho+bich+nga