It sure sounds like it is in limp mode to me, but why? It could be any of a dozen or so solenoids and sensors used with the transmission. Likewise, it could also be the transmission control module or the engine control module, itself.
Frankly, it seems more like an electronic issue to me than anything else. As you noted yourself, the Cube starts and runs nicely, it just won’t get out of its own way — limp mode.
To find out, if you can stand the slow-speed drive, take your Cube to a local dealership (yes they tend to be higher-priced, but they do have all the latest diagnostic software and that’s what you need right now). Let them know what happened and be sure you ask them, before you leave, possibly, to have a diagnostic scanner hooked up to your Cube and a diagnostic routine run — ask the service manager to handle this personally (you will get a hearing, believe me as dealerships are sensitive to customer satisfaction — it is a key metric. Most people don’t think this is true. However, it is, really!).
Be certain you wait for the results and I will bet you a jelly doughnut that you will find codes flashing up that indicate a sensor or solenoid has failed.
The reason I suspect it is an electronic piece is the nature of the failure. You drove the Cube 3,000-plus miles and in that time nothing happened. Then, you park at home and the next morning, bingo. If a transmission mechanical part had failed, you would likely be hearing the whining more loudly or there would be another indication such as slipping.
There’s a silver lining to this. The Cube is a great small car that should get you another 100,000 miles easily, so it is worth the $400 or so to replace and fix the electronic part. Let me know what happens. I am interested to see what the shop says.