"I also need to check the adjustment of the transmission kickdown (I don't think this would affect starting from a full stop).
When I got the car the trans kickdown didn't work at all. I got it to work by trial and error but I recently found the instructions how to do it correctly."
The TV LINKAGE has to be adjusted with a fluid pressure gauge (see correct year WSM). There is no ands, buts, ifs or maybes.
Also correct band adjustments.
It sounds like your problem(s) is mainly the LOAD-O-MATIC and the TV LINKAGE. You need to go to a DUAL-ADVANCE DIST.
Make sure all of the TV LINKAGE is not worn and any bushings are good. Engine mounts also. If the engine moves on torque application it will negate any adjustment.
The reason(s) for the single range in those years was fuel economy (2nd gear start) and to compete with the scrub POWER-GLIDE.
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