You need to scan for codes. Most probably there will be a code in the Engine and/or Transmission computer for no vehicle speed signal (probably "P0501 - Vehicle Speed Sensor A, Range/Performance" or similar). On your A4, the VSS is handled by the ABS module, so checking for ABS codes will likely get you closer to the actual fault. Perhaps a faulty wheel speed sensor. Once you make the ABS module happy, and clear all your codes, everybody else should fall in line. The automatic transmission does depend on accurate Vehicle speed signal for tuning the shifts. Your symptoms all point back to a problem with the ABS/Vehicle Speed.
Do note that just because there is a code for a wheel speed sensor does not necessarily mean the sensor itself is faulty. The sensor reads a "tone ring" (aka "reluctor ring") that is bonded to the CV joint. Rust buildup in the notches or cracks in the metal tone ring can cause faulty readings. Also, rust/debris buildup on the end of the sensor can cause issues as well. Because the sensor is reading properly some of the time, I would be more diligent about inspecting the tone ring and associated debris before purchasing any parts. It is possible to check the resistance of a wheel speed sensor with an ohmmeter. A further check is to actually see it's output. This can be done with an voltmeter in AC milivolts mode. An even better test is to connect it to an Oscilloscope (or a digital scope function of a scan tool, etc) and look at the quality of the waveform that the sensor is producing at various wheel speeds. It's not generally necessary to get quite that deep with the diagnostics unless the issue is proving difficult to rectify. Good Luck.
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