My suggestion would be to get a dedicated audio recorder such as the Tecsun ICR-100.
That unit has a built-in FM radio (which it can record from) and records to microSD with up to 32GB capacity. Tecsun has a very good reputation for very sensitive radios, so I would expect the same for its recording quality. Just a pity they don’t do DAB yet.
For recording DAB, you can connect your DAB radio to its line-in port.
If you record in MP3, you can use MP3DirectCut (link) to adjust the volume and split the recording up into individual MP3s, such as to extract an individual radio programme, remove the ads, etc.