I just bought the last three I needed and added them to my Plushie collection, but when I refresh the page, it still says I don't have the three I just bought and added! I even double checked with Account Search and it says I have them in my Plushie collection.

I got the A Pirate achievement yesterday and I believe I triggered it by going to the Plushie Collection page and then refreshing on the achievement page. It took a little bit for the achievement to come through but it eventually did.

Usually to trigger those kinds of achievements you need to click the "Show required items" button for the achievement on the achievements page. Sometimes there's a bit of a cache, so it may not trigger immediately. Hopefully if you go back though, it should.