Built-in spell checker not working in WordPress 3.3.1

Built in spell checker was always a bad idea especially since it relies on external services that might not support the actual language you are writing in. In addition those services might have changed their API since the days of 3.3.1 in a way that is no longer compatible with the 3.3.1 code.

Or am I resigned to installing another plugin?

IMO spell checking for 99% of the sites should be done as a browser plugin. The exception is when you have several writers and you want to force uniformity, but then you probably want to host your own dictionary, and will need a plugin for that.