... I shifted to where it felt engaged and then gently released the button on the shifter so it would stay in this "in between" position. There was no change in shift performance...
The "rev-up/flare-up/slip" happens when the trans has shifted into 3rd, but the torque is too much for the front clutch to grip. When you back off the throttle, that lower torque the front clutch can grip. According to the TSM, this can happen when the gear selector lever/rod adjustment is off, even slightly. I had this happen to me. I didn't check the TSM and it cost me the front clutch. The more your drive it while in this condition, the more you are wearing the front clutch. Wear it too far, you will have to replace it. If you have to drive the car before you get this fixed, I'd very strongly suggest that you use minimal acceleration when shifting from 2nd to 3rd, and when driving in 3rd. Don't even dream of pulling a trailer or carrying heavy loads.
The trans is pretty robust, so the most likely (and fortunately least costly and easiest to address) is the gear selection (P|R|N|D|2|1) lever/rod adjustment. Playing with the movement of the gear selector lever is fine, but the problem is likely due to play in the connections between the gear selector lever through to the input to the transmission: you can't get it into the right position and stay there - the play in the connections doesn't allow it.
If you or anyone before haven't moved the base house of the gear selector lever around, or adjusted the connection position on the rod that attaches to the bottom of that, then the most likely culprit is excessive wear to the connectors that connect:
* the gear selector lever to the rod (bushing, #19 in the diagram; rod, #21),
* the rod at the front end where it connects to the transmission (#25 in the diagram)
You'll likely have to find less worn connectors at the junk yard.
It's a tight fit to get your hands up there from the bottom. I don't recall if I was reaching around the skid plate (lazy) or if I'd already taken it off. I hope your hands and arms are small.
As noted in the Tip of my prior post, with new/different connectors, the correct attachment/tighten point of the gear selector to the rod may be slightly different than original. But the bolt (#24 in the diagram) that tightens against the rod will have either mulitple points on the end or a single point on the end with which to tightly hold the rod (have seen both types on Eagles). These points may have left indents in the rod that the bolt end will try to slip into, instead of tightening on the exact correct location the adjustment requires. Simply sand the indent(s) out. Do not accept an "off" adjustment, you will eventually lose the front clutch. A lazy transmission shop convinced me it couldn't be adjusted correctly and my front clutch was too worn by the time I took matters into my own hands and read the TSM and did the adjustment myself.
Eaglepedia parts diagram 38.23.1
The kick-down is almost always fine, excepting for the kick-down throttle linkage, which can be restricted from moving properly by accumulated grease, oil, dirt and rust. Brake cleaner. Here in the Rust Belt I always kept a spare ready to swap in. Rust is why the kick-down linkage is one of the must-grap parts when visiting a junk-yard. It's a pain to reach down in, but it's easy to replace and adjust (with the TSM, which you have on the way).
Note: I did encounter one gear selection lever assembly where the pivot of the selection lever was so worn that it caused so much play that adjustment was impossible and a less worn selection lever assembly was required.
It's a shame so many of the old threads are lost. This has been well documented in the past, as it keeps coming up over the years.