Relocating to Hampshire: opinions requested

[Fayelara]

We as a family of 3 plus dog we would like to move to Hampshire. We want to minimise our mortgage so decided the max will be 350k and I am just really worried that will mean we have to live in a bad area. Can anyone suggest any lovely areas popular with young families?
We love Winchester and would want to work there.
We need a min 2 bed with a big garden , infant schools and general nice vibe.
We love walks, the outdoors and we would like rural feel and comfortable commutes to work in Winchester.

You could look at Kings Worthy (a big village on the outskirts of Winchester). I know lots of people that live there and like it. South Wonston is another village slightly further up the A34.
Colden Common is the other side of Winchester and affordable.
Slightly further away is Chandler’s Ford and Fair oak.
Eastleigh is less desirable but I don’t really know why!

Suggestions - Grateley, Whitchurch, Overton, Andover on edges, Abbots Ann. I don't like Eastleigh. Look at school Ofsted reports, which give you a good idea of where would be good to live.