I noticed a trail from Shoreham to the North Downs
That’s the Downs Link, I think? It’s quite scenic at first, but then is mostly fairly dull. Doable on a road bike for sure in the dry (doesn’t get anywhere approaching ‘technical’ until the ND) but it may be a bit grim at the moment. There’s lots of fairly quiet roads north of Brighton which I’d personally prefer to do at the moment
When it’s a bit drier you should definitely investigate Stanmer just outside town. Lots of purpose built trails. Not great in the wet, but when it’s a bit drier I’ll be happy to show you about and lend you a bike if you want to “try-before-you-buy”?