I have a honda jazz with an automatic transmission and im experiencing some hesitation when accelerating with the AC on, but with the AC off it runs so smooth that its like im riding a different car. And ive observed that when parked or in a stop with the engine on the car shakes/vibrates for like 2 seconds and it shakes quite often but for only 2 seconds then goes back to normal. And it also runs very rough when the car is shifted to drive with a cold engine, the rpm drops below 500 and hesitates bad with a weird engine sound but as the engine gets warm the roughness disappears along with the weird sound.
what causes all these? i can imagine it involves a lot of things but can you guys help me out?
Ive also observed that every time i let off the gas, the rate of which the speed reduces varies from time to time, it sometimes feels like downshifting in manual car as i left of the gas(that slight braking feel when u downshift), and sometimes its smooth. is it a normal car behavior?
Edit: there are no codes and no check engine light