Only first gear, everything else is "neutral"

Hey 2CarPros Community.

Yesterday my van suddenly broke.
We drove ninety miles on the highway, and when we got off at the first intersection it appeared:

Only the first gear works, when it tries to shift to second gear the transition gets stuck in "neutral". It is actually not quite neutral, I guess the transition get stuck between the gears or just could not get into the second gear. You can hear the gear wheels "ringing" a bit at that point.

And when that happened, nothing works any more. It stays in that "intermediate state" and also will not got back in first gear after stopping the car. So, I have to go back to park (park still works), turn off the engine, turn it back on, and then it will go into first gear again when I dial drive, "3" or low.
And the same happens again when I start driving again.

Reverse gear also does not work. Even after start up.


TCM Error Codes:

PO740 TCC Control Circuit Error.
PO734 Gear Ratio Error in Fourth.
P1738 High Temp Operation Activated.
P1790 Fault Immediately After A Shift.
PO0732 Gear Ratio Error In Second.




My suggestion after a short research:

If the transition oil filter is clogged, there will be no oil pressure any more to allow shifting, right?
So, we did a first short test with a OBD scanner. The shifting (electrical output) from the computer works (so the gear indicated on the scanner).

Is my suggestion a reasonable possibility what might be wrong and causing the problem?
Or do I have a completely wrong understanding of how the transition works?

We bought the van in February and drove about 4,000 miles since then. We have not change transition oil and filter in that period.
We also had issues with the TCM and limp mode one month ago, and used the car for about four weeks in limp mode for just our five miles way to work. So, it was used with a significant increase of RPM's recently. And I thought that would fit into the "story" since higher RPM's causing a higher flow through the oil pump as well, and I do not know when the last transition oil change was made.


Thank you for your help in advance!
Otherwise, it will be the end of this van for us, because it is stuck two hours away from home in Green Bay right now.