Kind of does sound like the solenoid is stuck. I've never had to do it and don't recall discussion here, but the bypass seems a good option. You only lose being able to hold or force a downshift to 3rd. IPD also has a bypass kit and cheaper than the Toyota one you found on eBay.

These AW71L trans are so reliable, even on a turbo, and it's amazing we don't have more issues like this at these high mileages. With that in mind I'd start with a good flush before going to all the effort of doing the bypass. You don't even have to crawl under if you want to skip the draining, just do a longer flush using more fluid and make sure you periodically run it through all the gears. I presume you checked the current fluid level while running. Have you had to top up in recent years? Has trans fluid been dripping on the ground? When was the last time it had a really good flush? Drained first? Synthetic? Shouldn't be it, but check for a sticky shiftdown cable just for the heck of it.

I recently did a pan gasket and it drained for over a week while I was working on trying to get the fill tube disconnected (had to resort to cutting it, should have done that right away, so simple and easy, HomeDepot for the brass fitting. ) I was surprised how clean the filter and pan were not having been flushed in well over 100k and the filter hasn't been cleaned in probably 300k (km). Switched to a good synthetic and shifting is as good or better than ever. I've been dealing with oozing fluid since the day we got that car and when tightening the pan no longer helped I finally dropped the pan, no gasket, just sealer, slowly going mushy. For some reason FCP was pretty much giving away good Corteco cork gaskets at around $2 each so got a spare for the other 940. All 100% now and a new trans mount bushing in the process that was starting to go. Now have to do a front mount with parts now on the shelf waiting for me to get round to having the time in warmer weather.
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Dave -still with 940's, prev 740/240/140/120 You'd think I'd have learned by now