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How long has it been since the bands were adjusted last?
The bands need to be adjusted at every fluid change, every 30,000 miles.
If you haven't had the bands adjusted in at least 7 years then the first thing you're going to need to do is attempt to adjust them and see where you're at. The kickdown band is the forwardmost band and is applied for second gear, your problem involves the kickdown band in one way or another.
It may be that the band is out of adjustment simply from lack of periodic adjustment.
The band anchor may have fallen out of place. The anchor goes between the band and adjuster. It can fall out of place because it breaks or if the band gets worn far enough without proper adjustment.
It could be a leaking or damaged kickdown band servo. The servo is the hydraulic piston that applies the band.
It's also possible that you could have a bad output speed sensor or governor solenoid or transducer issue. If the check engine light isn't on or there are no fault codes stored then these wouldn't be nearly as likely as a band issue.
It's something you can do if you are confident enough to tackle it and have the tools. The kickdown band is adjusted from outside of the case. Look near the front cooler line and you will see the band adjuster. You will loosen the lock nut and then torque the adjuster in the center to 72 inch pounds. The adjuster is then backed off 2 7/8 turns and the lock nut tightened back down.
If you find that the adjuster threads into the lock nut all the way an won't tighten down then the band anchor has fallen out of position. The next step would be to remove the pan and look for it, and then the valve body if necessary. I see that you've already had the pan off, so if the anchor has fallen out of position then it would have to be on top of the valve body if you didn't see it in the pan.
When you backed off the lock nut and then tightened the adjuster and torqued it to 72 inch pounds how far did it go in? Did it bottom out in a few turns? Or did it thread all the way into the lock nut without bottoming out?
The band isn't adjusted then. If it threaded all the way in then it wasn't possible to adjust. Adjusting the band requires it to be torqued to 72 inch pounds and then be backed off. If you couldn't get the adjuster torqued down then the band isn't adjusted.
If it threaded all the way in then the band anchor has fallen out or the band is worn beyond use. If the anchor is out it is either broken which caused it to fall out or it fell out due to lack of periodic band adjustment.
Remove the pan and look for the anchor. It's roughly one inch square and somewhat of a wedge shape, it has a notch in one end and a concave part on the other end. The anchor goes between the band and the adjuster on the outside of the case. If you don't see the anchor in the bottom of the pan take a look and see if you can see it between the band and adjuster. If it's not there then the valve body will need to be removed to look for it, it will often fall on top the valve body.
Have you checked for fault codes?
Any of them should work fine just for reading codes. If you're interested in looking at actual data and such then you might look for something with more capabilities than a code reader, but you'd spend much more. A basic code reader will start at about $50-70 and up. Most parts stores will read codes for free in the parking lot too, which may be an option for you.