I understand your concern. There are no internal screens to clean but the pipes under your sink can be cleaned. Based on your symptom, the pipe is likely restricted where the two drain pipes Y together from the disposal.
If water drains properly after running the disposal then there was likely food debris in the disposal because of not running the disposal. If you have a double sink and you have the stopper in the other side, it can also cause the other side with the disposal to drain slow. If the vent pipe is stopped up it can also cause water to drain slowly.
If the other side of the sink drains okay and it's just the side of the sink with the disposal that drains slow then you likely have a restriction in the drain pipe where the two drain pipes Y into each other. The drain pipe will need to be taken out and the blockage removed.