Answer the following questions:

-Is your internet browser up to date?-
Uhm…no–Then go update it! Restart your browser, then see if that fixes the problem.
It’s up to date, smartass/Updating didn’t solve the problem–Alright, go onto the next question.

-Are you using Firefox by chance?-
OMG how did you know?!–I’ve read few posts complaining that the new Tumblr audio player isn’t compatible with Firefox. Try using a different internet browser or revert back to the most stable version of Firefox, if possible.
I don’t use Firefox–Onto the next question then.

-Do you have Missing e installed?-
Yes because YOLO–Try uninstalling the program, restart your browser and see if it changes anything.
I don’t have Missing e/Nope, still doesn’t work–Next question.

-Have you tried User Agent Switcher?-
What’s that?–It’s an awesome add-on that allows your change your current  browser’s user agent to different browsers. Try installing it, switch it to an agent like IE 9, refresh your window, and see if anything changes (note: this will cause you revert to the old posting layout).
Tried it and didn’t work–Then I’m all out of suggestions. However, there’s some cases where you can listen to the audio on the site itself as oppose to the dashboard. You may have to resort to that until Tumblr fixes these “glitches.”