If you are open to inland market towns, can I suggest Skipton, North Yorkshire.
No sea, admittedly but plenty of stunning views, and it has a lovely swimming pool. Next town along (Ilkley) has an open air lido for swimming.
Beautiful countryside, very close to the Yorkshire dales, loads of walks straight from the town, walking along the canal is lovely, it runs right through the centre of town.
Loads of amenities in the town, great library, cinema, pubs, restaurants, cafes, independent shops. Great sense of community, lots of events throughout the year.
It’s a popular destination for tourism (and has a fantastic castle) but it doesn’t get deserted in winter or particularly overrun in summer, just a buzz all year round.
Well-connected -direct train service to London (I think at least one per day) plus frequent direct trains to Leeds, and Bradford, and the historic line to Carlisle, easy to get to Manchester too, and the Lake District less than an hour away. Direct train to Morecambe for a taste of the coast.
You could get a little stone Victorian terrace for well under budget, or something more modern with a garden would be affordable too.