My esc off light and check engine light both went on while I was driving and the car barely moved. What's could be wrong? I also had a problem around ten months where my check engine light went on and my car was shaking vigorously, but the next morning all was fine. Both of these problems happened after a rainy day where it became colder overnight. Could these problems just be related to the weather?

I had this exact same problem...turns out it has to do with a recall having to do with the stop lamp switch. I took it to the dealership and it only took about 20 minutes form them to fix it.
Hyundai is recalling 531,894 MY 2006-2007 Accent, Entourage, Sonata, MY 2007 Azera, Elantra and Santa FE, MY 2005-2007 Tucson and MY 2007-2008 Veracruz vehicles. The stop lamp switch in the affected vehicles may malfunction. A malfunctioning stop lamp switch may cause the brake lights to not illuminate when the brake pedal is depressed or may cause the brake lights to remain illuminated when the brake pedal is released. A stop lamp switch malfunction may also affect the operation of the brake-transmission shift interlock feature so the transmission shifter would not be able to be shifted out of the park position. It may also cause the electronic stability control (ESC) malfunction light to illuminate, and it may not deactivate the cruise control when the brake pedal is depressed. Any of these malfunctions, alone or in combination, may lead to a crash. Dealers will replace the stop lamp switch free of charge. The recall began on April 23, 2009. Owners may contact the Hyundai Customer Assistance Center at 1-800-633-5151. This recall was the result of a Preliminary Evaluation conducted by the Office of Defects Investigation.

I had the same problem, took the car to the dealer, the said it was humid in one of the control boxes. Claimed to have cleaned it out and car was good for 2-3 months. Now just came back :(