That is indeed an A604. It goes by a different designation today but the design is the same.

Staying in second gear is "limp mode". That allows you to drive slowly to a repair shop without needing a tow truck. You already learned that you have to turn the ignition switch off to get it out of limp mode. Once it defaulted to limp mode, there will be a diagnostic fault code stored in the Transmission Computer. Having that code read will identify the circuit or system that needs further diagnosis. In general, if it starts out right away in second gear, it is usually a sensor or electrical problem. If it goes back to second gear during or right after a shift, typically an up-shift, it is usually due to slippage in a clutch pack.

The problem with slippage is years ago that caused sloppy or mushy shifts that gradually got worse over 100,000 miles and we got used to it, then we'd marvel at how crisply the transmission shifted after it was rebuilt. With Chrysler's design, the computer detects how much fluid it takes to apply each clutch pack so it knows how much wear has taken place in each one. Every up-shift has the timing modified to overcome that normal wear and make it undetectable to the driver. That means it might engage third gear a little sooner before it releases second gear. The problem is you have no way of feeling that sloppiness or knowing it's occurring, . . . until the day it can't update any further. Then slippage occurs which sends it into limp mode.