Have you looked at Deal?
I know a family with now grown children and an older married couple who moved there from this area more than 10 years ago and really like it. Nice high street with quite a few good restaurants.
For a city, I like Bristol a lot. Less than two hours by train to Paddington. Good hospitals. Easy access to countryside - Cotswolds, Devon and Cornwall.
Also, look at Wimborne Minster, Dorset, a historic market riverside town north of Bournemouth. I did a lot of research a few years back and WM ticked a lot of boxes for me.