Have you thought about North Herts. Straight down to Hatfield on the A1, or by train.
Hitchin is probably most expensive, though you should be ok on your budget for a 4 bed. Lots of great individual shops, character, and lots of nice restaurants.
Letchworth & Baldock are cheaper. Shops/restaurants not quite so good though there are hidden gems in both.
Royston, right on Cambs border is where I live. You would definitely get a good 4 bed on your budget here. We have a 4 bed, admittedly town house style, with study too and paid a lot less than you're proposing 2 years ago. It'd be about 25 mins on the train, 25-30 mins on A505/A1 now Baldock bypass is open, down to Hatfield. The shops here are nothing to write home about but it's only 20 mins to Cambridge (again by car or train) and there are lots of nearby villages with good restaurants/pubs.