Sounds very similar to the issue I have been tracing in mine for the past week or so. See the link below for what I found, turns out I have a bad solenoid.
When you are having trouble taking off have you tried selecting the 1-2 range on your gearbox, this is what I found with mine, once warm the auto would try to take off in 3rd gear, but if I put it in 1-2 it uses 1st. From my reading so far I suspect that the position of the shift rod in this position allows the gearbox to activate 1st gear without the aid of the solenoid.
XJ Auto transmission issues and diagnosis