From what you're saying, I think it is definitely a transmission problem because my car although its an 2004 Acura MDX, had almost the same problems.
First it started to hesitate, but I thought it was just the transmission fluid so I didn't do anything about it for a couple weeks and then it started to get a lot worse. Such as jerking back strongly every time the gears would change, and struggling a lot.
Finally, we took it to the auto shop and sent us to a transmission specialist and we wound up getting a whole new transmission costing a couple thousand dollars.
It sucked.
**Tip: Don't go to "Lee Myles" (Transmission Services) though if there is one nearby you. They are absolutely THE WORST. Although they were good at fixing the car, they are rude, has incompetent service, and tries to cheat people out of money.
Anyway, hope this helped, and you should get the car checked before it gets worse like mine did