Toyota - Corolla :: 2010 - Makes Vibration / Shaking While Getting Uphill When Gas Is Pressed?

My Corolla was purchased new, and at 29500 miles it started getting a strange vibration (not the steering wheel but the car itself) whenever going up a hill or at higher speeds on the highway but ONLY when the gas was pressed (You once you let your foot off the gas at the same speed the vibration stops). After three attempts of taking the car to the dealer where the vehicle was purchased, the only responses I got were "We cannot duplicate the problem." After about the 6th time, I drove the car with one of the Sales Managers because the Service Manager wasn't there, and that manager felt the vibration, the dealership kept the car for three days, after which told me nothing is wrong, and returned the vehicle to me. I then turned to Toyota Headquarters, who set me up with a case and a case worker, and all he told me was to try working with another dealer. I went to a second dealer (closer to me house), and they did several inspections and doing a rotation, balance, and alignment.

They said the problem was unknown, but that they RECOMMEND that I get four new tires, thinking that MIGHT solve the problem. I bought 4 new tires, and after getting the tires replaced, and balanced and aligned AGAIN, the problem continues. The Toyota dealership then told me that they don't know what the problem is, and now I'm out of options. I don't know where else to turn, who else to speak with, and what else to do. I already paid off 2 years of the finance on my car, and have 2.5 years left. Side notes (whether irrelevant or not): The day I bought my car my mother got the same exact car just a different color. Her car now has 20000 more miles than mine, and runs beautifully. My car on the other hand, besides the vibration issue, seems like it has less power and has a very difficult time accelerating, unlike the other Corolla.