It's a magnetic (or MM) cartridge. It could be faulty -it does happen.
I believe on the later sliders the contacts lay flat, I think the earlier ones (or I may be getting muddled with the BSR ones) had a slight 'bump' to help contact.
If everything with the arm, head-shell, slider and wiring have been checked and double checked then it is most likely the cartridge or the turntable input stage on the amp. The main amp must be ok for the radio to work.
The only real option is to try another cartridge - do you know anyone with an interest in hi-fi who might have a spare to try? Needs to be a magnetic one - not ceramic.
It might be worth checking the connections on the underside of the deck, the small wires from the arm will either be soldered to a tag strip or less likely fitted with rca/phono plugs and in turn connect to the amp, this will involve some 'exploring' to open up the cabinet to access the underside of the deck.
But the fact you are getting sound, but distorted suggests the cartridge or amp.
The Sonotone cartridge can still be bought £25 -£30 found it on ebay and at Musonic, a quick google will bring some links up.
An alternative would be the Audio Technica AT95e, a very popular budget cartridge on the forum and around £30 -£35 or there is the cheaper AT91, about £20.
Obviously if it's not the cartridge then there will still be a repair bill to fix the amp - but make sure you get a quote if you go with that option, it shouldn't be overly expensive. But you'll have to judge what's worthwhile.
But if you can 'borrow' a cartridge that would be the first option to try.