If CTIS is off and the truck is sitting, idling, once the tanks are aired up the compressor should only cycle once every 10 or 15 minutes. If it's once a minute, you have a big air leak. The spec is for it to kick in at 90 PSI and shut off at 130 PSI.
If the dryer is fouled up (often seen in A3's with air leaks), you will have moisture and snot when you drain the air tanks. If the dryer is working right, the 'wet' tank is bone dry all the time - Always.
In the A3 CTIS system, all tires are connected in parallel. There is no provision for monitoring or separately filling individual tires. The tires will all reach equilibrium at some common pressure. If that pressure is less than the set point, CTIS fills them. It if is higher, CTIS vents the excess. If the vent rate exceeds a set CFM, the wheel valves on each wheel slam shut to isolate the tire from the system to avoid running the tire flat. When not operating, all the CTIS plumbing is purged to save the wheel hub seals and to force the wheel valves to close & isolate each tire.
When the system starts up, the CTIS 'puffs' a short burst of air into the system to open the wheel valves and let the tires charge the lines while it monitors the pressure. This is the puffing sound you hear when you turn it on. It puffs and sees if the line goes above 20PSI. If it doesn't get there, it tries again and measures. It will repeat a few times and if no joy, displays 'Flat' on the screen. All this means for sure is that the system could not get the lines to charge over 20PSI either due to plumbing leaks or the tires were less than 20PSI.
If there is oil in the system, it will gum up the valves in the CTIS manifold system (under the truck, aft of the battery compartment). It will work its way to the axles and the wheel valves over time and gum them up too. A bad axle seal will put gear oil into the wheel valves for sure. This is why people were asking whether the oil was thick/thin/had distinctive gear oil smell, etc. This is a way to easily diagnose the oil source.
The comment asking if you have checked your axles to confirm someone didn't fill them with 10w motor oil is a very good question and more important than the CTIS consideration at this point until you can confirm. Light crank oil will destroy the axle & Diffie.