1966 Plymouth Valiant Shift Linkage Help

It sounds like the bushings in the linkage are warn. This allows the linkage to reach neutral, but the trans is still in the last gear selected. The interlock in the side cover won't let you get into two gears at once. you would have to move the arm to the neutral position to free it up.
I don't recall the bushings, but I think they were a rubber-type material. Metal would be better.
I put a Hurst 3-speed shifter in, but you must fab a floor console because the linkage is inside the pass compartment.
I had a '66 V200 with a 170 & a 3speed. I used the V-8 tail pipe & a Thrush 2" muffler & Accel points in the dist. That got rid of the flat acceleration when you rev it up in 2nd & shift to 3rd.
I liked that so much that I found a '66 273, 3-speed 2-door That's when the column shift linkage became a problem. I bought a Hurst Syncro Loc shifter, which mounts on the floor inside the car(the rods are above the floor) but soon fried the clutch.
But I digress. Check the bushings in the tranny arms, & replace.
SpeedyEd