Problems with accelerating
 

I have a 2006 S-Type 3.0 with 108k miles, and its having some issues when accelerating from low speeds. I'm not sure what it could be, I'm not very good with cars. When I press the gas from a stop or low speed the car will hesitate and the rpms will drop. I can hear and feel a rhythmic clunking a few times before it feels like something catches and the car starts moving and the rpms slowly climb back up. It seems like it happens a lot longer and worse the further I press the gas. Sometimes the drop is so sharp it even makes the car slow down a good deal. At first I thought it was the transmission slipping, but if that was the case then the rpms would increase and not decrease right? I haven't noticed it at higher speeds, only in the first 2 gears. Before this started happening it would sometimes jerk forward really hard when I quickly gas it at a certain point after slowing down, right about when it would downshift. There's also been a slight delay when shifting into reverse and drive, about a second or two. I tried changing the transmission fluid but that didn't help much. I'm not sure if I put enough in though, I drained about 4 quarts out and put another 4 in but from what I've found online apparently it's supposed to hold more like 6 or 7 quarts? Since then it's been happening on and off, sometimes it'll go a day or two without doing it once and then one day it'll do it almost everytime. I don't really feel comfortable bringing it in to a mechanic until I have a good idea of what the problem might be. Any help would be greatly appreciated.

See attached. If you aren't sure what's wrong with the vehicle, take it to a Jaguar specialist as guessing can get extremely expensive.

I'm not that familiar w/Jags just yet but I have 30+ yrs working with cars.
At first your description matches low fuel pressure. But then you say it'll jerk fwd. So w/o seeing it myself I'm not sure which to pursue first. Fuel or tranny.

Based on your description here is what I'd check & do:

1. Look into low or intermittent fuel pressure. When was the fuel filter last replaced?
2. Make sure tranny fluid is at proper fill level.
3. Disconnect battery and touch cable terminals for at least 15 seconds.
4. Reconnect battery and put car through all learning procedures. (KAM is cleared and it must now relearn shifting. (commanded times vs what is really happening will be adjusted by the computer))
5. Unlikely but maybe an intermittent and gross vacuum leak,

At 108K miles, your filter (contained in the transmission pan) and connector sleeve should have been changed as well... What kind of automatic transmission fluid did you use in the change? The wrong ATF can cause all sorts of shifting problems too...