My Aero T96 used to jam and be resolved just as you describe, shifter "box" fastened tight and seemingly in proper adjustment. Eventually one of the synchronizer snap rings broke, (a 14 cent part at the time) and the trans locked in 3rd gear, resulting in decades of being parked........ with the new snap ring in the glove box.
The grandsons & I completely disassembled the trans this summer and all the brass parts were worn, one ring to almost nothing. If your checks don't show anything loose you might want to take the cover plate off the top of your trans for a look. Don't know about a J but on an Aero there's an access plate in the front floor for that.
On a related subject, after buying gear oil for my "new" transmission & OD, I reread the BW overdrive manual and noticed it said to use GL-! rated oil, the modern higher number GL oils apparently have additives that eat up the brass parts in older transmissions. I found the correct lube at NAPA.