SOLVED
I just put on the Emergency Pirate Outfit and then removed the buff from that, and then everything was back to normal. 
Leaving this in case someone else has the same issue, tho I really feel like I should have thought of that earlier. 
Nvm, I died and now it’s back?
Pirate costume from previous event was stuck.
There was no buff, using toys, dying, relogging and so on didn’t work.
I haven’t actually seen it, but does it give you a buff that you can right click to deactivate at all?
Nope. 
Some people have reported that joining an arena skirmish removed the costume, if you wanted to give that a try?
Edit: Just another thought. I wonder if the Savory Deviate Delight might reset it? They are quite cheap to buy from the AH. It’ll give you the appearance of a ninja or pirate, but you can right click the buff to remove it.
Was also stuck with the pirate outfit untill I joined a arena skirmish so can confirm that it does indeed remove it.
Didn’t work either, but I’ve solved it! Thank you tho. 
When I signed up for deck-scrubbing duty, no one warned me I would be stuck on the ship for life. Send help. I mean yeah haha being a pirate is great, yarr harr!
What didn’t work and how did you solve it?
Don’t be negative about it. I was also still a pirate when I logged out and I hope to still be one.
The solution:
YARR
Yar har, fiddle di dee
Being a pirate is all right with me
Do what you want 'cause a pirate is free
You are a pirate!
I can confirm that joining an Arena Skirmish works.
Pirate Free since 10:33 
Joining an arena skirmish works.
Oh no, now I can’t do any skirmishes or I will lose my pirate costume.
Please Blizz make it perma 
I just switched to Assassination and I was no longer a pirate.
Yeah, some of my chars lost the costume surprisingly early, and I suspected it has to do with spec changes.
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