*As of 3/21/19, it looks like Playground has addressed & fixed the problem, and disconnects aren't happening anymore. So if you were putting off doing another attempt, you should be good to go now. *
In the main menu, under the section "Online Adventure" is a section for "Quickplay Adventure." Select it, and then click on "Free-for-all Racing." Its basically the same as Team Adventure, except you're solo and you will always play out all 5 rounds. Once you complete all 5 rounds, have gone through all the ending screens, and get back into a lobby, the achievement should pop.
On my first attempt, I received a disconnect error during the end scenes after I completed all 5 rounds, and was booted back into a lobby. This didn't count for the achievement, even though I did technically finish all the races. So if you're having problems getting the achievment even though you completed an adventure, you may have just gotten disconnected early.