Hi, I am retired and live in the New Forest. I suspect it depends a bit on what your thoughts are on what sort of home you want to have. You could get a retirement flat in Brockenhurst on your budget, or, just about, a small house in Lymington, or a slightly bigger place in New Milton.
All these have a station, Brockenhurst has very good connections to London, about 90 minutes to London Waterloo, handy for the theatre etc. Your budget might well go further if you look at villages without such good transport connections.
Lots of people retire here, they are very friendly communities full of lively people interested in the arts, keeping fit, all sorts of hobbies and activities locally. See what is going on the the local village halls/ community centres.
Lymington and New Milton have good shops, Lymington also has a Saturday market. There are quite a few independent butchers/ bakers/ greengrocers etc locally.
The surroundings are of course very beautiful, and the sea is close by. Being in a National Park has its downside if you want to extend your house! but it is lovely to be so close to so much beauty - and I think it doesn't matter so much if your garden isn't huge as there is so much nature on your doorstep, and the ponies helpfully keep the grass short etc.
Bournemouth and Southampton have big hospitals, a variety of shopping, big theatre and music venues.
Christchurch is indeed also very beautiful and has a lovely feel to it.
It would be a good idea to come down for a weekend and have a look around. That was what we did, and my husband took one look and was determined to move here...