So after about a minute of driving, the CEL comes on, and stays on until the car is turned off. Though, everything seems normal and smooth, and nothing changes when it comes on.

Diagnosed it and it shows:

22 - Crank angle sensor system
23 - Top dead centre sensor system

Any ideas?

When the CEL is on, turn engine off and try to restart. A 23 will not allow the engine to start while the fault is active, but it tends to be an intermittent error.
TDC sensor, or cam angle sensor, is inside the distributor.