The specific description of the PO601 code is "internal control module fault condition (check sum) detected." There are indeed several controller modules on your vehicle. The issues you have mentioned do sound like PCM issues; but this is not 100% certain. Here's what I would suggest. First of all; make sure that there is an auxiliary ground cable between the battery and the vehicle body (or one between the motor and the firewall) in addition to the main ground cable from the battery to the engine block. If either of these cables are missing, corroded, loose or damaged, or bolted to a painted bracket; they must be rerouted to an engine bolt, or repaired or replaced. Then, have the alternator output tested with an accurate voltmeter and an oscilloscope (many tune up and electrical shops know how to use oscilloscopes for alternator testing). What I'm concerned about is whether there might be a diode out in the alternator; this could give a normal voltage reading on a voltmeter, but would display a very unstable waveform on an oscilloscope. And such a waveform could destabilize the PCM. After the alternator has either been pronounced good, or replaced; try disconnecting the battery for 5 or 10 minutes; and turning on all the electrical switches on the dashboard while the battery is disconnected. Then turn all the switches off before reconnecting the battery. This should clear any abnormal memories in the computer. It might also be worthwhile to clear the codes with a code scanner at that time. You can expect the car to run strangely for the first few minutes, as it relearns the normal input patterns. If codes return; they indicate unresolved problems.
If the checksum message returns; then I would see whether the transmission controller could be eliminated as a posible source. Another option would be to try reflashing the PCM. That's about all I can contribute from here.