Most likely the bearing that the drive unit pivots on is bad. It causes the drive unit to twist when plugging, which causes the fan on the drive motor to hit the side of the case and lock up the drive motor.
There are usually metal shavings at the bottom of the motor from the fan visible.
There will not always be shavings. Yank the drive unit and check the bearing.

one of the very first checks, is to use an ohm meter and check for "shorts to frame" on all the power cables (must be more than 100k ohms resistance between and power cable and the frame, with the battery disconnected and out, not touching the truck, if you have less resistance between the frame and a cable, trace and fix that before you try further troubleshooting).
Also check the battery for good voltage at rest and under a load (hydraulic stall at full lift is about as good a load as possible). I was suggesting a 'quick and dirty' way to consider if the battery has [with age] gotten to a spot where -it- is a/the 'short to frame'. you would not be checking between battery positive and negative, but between the frame of the battery and one then the other of the power cables.
electric forklifts for the most part (and all ev1 controller equipped trucks), do not use a frame ground, but rather have a completed circuit back to battery negative. both in the control circuits and the power circuits.
is there a backup alarm on this truck? is it correctly wired, both positive and negitive?

It is an EV1. I think I'll have to go try a drive and see. It's pretty much always been this way so I'll have a hard time recognizing the difference. I may take a video.

Are the rectifiers serviceable or are the some sort of surface mount on the EV1 board?

EDIT: After a drive, I could not reproduce the symptoms but I have no load to haul. As this typically only happens with a load, I am not going to put too much stock into this not happening.

With the brake disengaged (with the floor pedal) and driving forward, I 'slam' the joystick into reverse and it slows quickly (is NOT abrupt) then changes direction. The same with driving reverse and putting the joystick into the far reverse. So I am thinking that rectifier is A-OK.

Can you read the part number of the control card, EV1, EV100, or EV100LX. On the older EV1 and EV100 if the #4 rectifier is going bad it will cause stiff plugging ( when you change directions the machine will be very abrupt and not come to a smooth stop and accelerate away smoothly)

It wants to keep going fast but I generally release the joystick (controlled) to center when this happens.

When this happens, does the machine continue to want to full speed in reverse or does it just go "bang" and stop?

I say that it only happens in reverse because in forward, I have never had the issue unless I am trying to go full speed and put the joystick full forward. The reverse thing happens during non-WOT (so to speak) as well as WOT.

I can try a 12V bulb but it runs ~36V soooo maybe I shall try a 36V bulb?

I have voltmeters, clamp-on DC ammeters, etc etc which I'm sure can help further

my suggestion was not so much clean the battery, but the area of the truck's frame directly underneith the battery. do you still believe the concern is ONLY in reverse? can you light a 12volt test light by attaching one end to the battery frame (metal case) and the other end to either (try both) end of the battery cable?

It's very hard to read the serial number but I want to say it's 1A111975 (the last digit is very hard to read).

I see no metal shavings at the base of the drive motor.

I can be maintaining constant pressure on the spring-return joystick and it will still happen so something must be sensing the speed/joystick position and want to kick it into high gear (for whatever reason).



EDIT:
In terms of the batteries, we have cleaned the batteries and have an amazing charger for them. We perform maintenance on the batteries more than anything else in this shop. When we purchased the machine ~5 years ago, we swapped all the batteries from bran new ones and had our electrician install them (he is a VERY thorough guy).

We just checked the water (all cells were good) and did a gravity test (also good). We always clean the terminals and if there is corrosion, they are either changed or cleaned right away (stop usage). All connections are good and the batteries still hold quite a charge.

This truck might have the directional switch and the high speed switch stacked on top of each other if it is a very old one hard to tell without serial number. also big question would be if you have metal shavings at base of drive motor. then it could be steer bearing between motor and drive unit

Serial number always helps get better guesses and opinions.
Has anyone pulled the battery out and cleaned all the corrosion that builds up under the battery and replaced any power cables and connectors that seem to be soaked in corrosion ?
How clean is the the battery, both in a housekeeping manner, and in the 'electrical shorts to frame' manner?
I would also want to know if they checked the battery at rest AND under hydraulic stall load to see if it is good. (what were the numbers of the results)
Next, I would have the directional contacts replaced (as a complete assembly, not just the tips), since you say, "forward acts as expected", and the only differences between forward and reverse are in the contactors control of the electron flow direction.