It's peculiar because the 420SEL starts off in second. You have to really mash the pedal from a dead stop to engage first or start off with the shifter in 2 (or B). Then you'll feel three upshifts.

As to the mushiness, could be a sign of big trouble or the transmission vacuum modulator on the left side of the transmission housing could use a tweak. These transmissions are sophisticated but you can try this - remove the black rubber cap over the center of the modulator, pull out the T handle enough to turn it, give the T handle an eighth turn clockwise, push in the T handle, put back the rubber cap, test. Does that firm up the shifts? The proper way to set the modulator is with transmission pressure gauges so don't go nuts with adjustments.

Sixto
95 S420
87 300SDL