Transmission Shifting Problem hesitating in shifting gears second to third and third to fourth

I have only had this vehicle about six months and am a newcomer to the "joys" of driving (and repairing) a Jaguar. The first three months of ownership/driving (one hundred miles/week average) were completely fine, uneventful.

After that, however, I noticed that occasionally (maybe one shift cycle out of twenty) the transmission was a bit hesitant when shifting from second to third gears and/or from third to fourth. While the rpm's were higher than they had been previously (now shifting around 3000 instead of 2000 to 2500), it was nowhere near redline, just significantly higher than I was used to with my old Granny-Style PT Cruiser! Still, more often than not, it drove just fine.

The last six weeks, however, it is shifting late like that almost three quarters of the time or more. Again, the rpm's are not any higher than maybe 3500 but it is clearly having issues. This car has the "Sport Shift" button/controller that, when engaged, is supposed to squeeze out every last drop of power it can before shifting--i.E, waiting until probably about the neighborhood of 3500 rpm's before shifting, though this feature was not engaged.

A couple of times, however, when it seemed like it just was not going to shift at all, I have noticed that often, if I engage/disengage this sport shift button (sometimes more than once) it seems to force a shift each time (both engaging and disengaging) and this makes me think it might be more likely a problem with the Gearbox Controller Unit/Module rather than the transmission itself.

Of course my first thought at the very first sign of trouble was to check the fluids unfortunately, I felt pretty stupid after spending more than an hour searching for the dipstick before consulting the manual and discovering there was not one. The only way to check the fluid level is to get the car on a lift, and basically pull the fill plug and then fill it to the point of overfill. I have not done this yet and likely probably should but several decades of "Murphy's Law" where cars are concerned, it seemed more prudent to start looking for the replacement.

Meanwhile a couple non-mechanic know-it-all's (said with love) have told me that in their opinion it is more likely if not most likely to be clogged filters and that I should shell out the $350.00 or more for a full transmission service. (When looking at an $850.00 used with warranty transmission replacement though, $350.00 seems an expensive gamble.

Last but not least, I have most recently discovered that it seems to operate almost completely normally if I drive it like a man of twenty five rather than a man of forty five not necessarily hot rodding it, but not doing the pleasant, near zero-g accelerations men of my age seem to do, but rather driving it like it was a Jaguar. Logically, this puts those rpm's at-shift numbers naturally into the 3500 range so it could just be coincidence, but I still wonder, is this all my Jag's way of saying, DRIVE ME!