I live just outside Abingdon and my advice to anyone thinking of living anywhere near the A34 or planning to live anywhere where they will need to use it a lot is: don't.
The A34 is a nightmare of the road, especially, if like us, you live off the stretch or have to regularly use the stretch between the M4 and M40 (which you would, Sabresprint). The A34 is dual carriageway and the main route for lorries between Portsmouth and Southampton and the Midlands. It is heavily overloaded and not a day goes past without accidents and holdups. The reason it has never been turned into a motorway is because there are no alternative routes between the M4 junction and Abingdon, other than small, windy and much longer country roads.
Yes, Abingdon, is cheaper than Oxford, but anywhere short of Central London is cheaper than Oxford-within-the-Ring-road. Newbury. too. is expensive. 3 bed semis in Whitchurch, or indeed the whole of Oxfordshire/ Berkshire area cost £300,000 and over. Hampshire is as expensive in many areas.
Whitchurch is being overtaken by new building but it is a very pretty small town within a very attractive area of countryside. It is equidistant between Newbury and Winchester (both via the A34) and Basingstoke is accessible only by windy country roads, so shopping outside the limits of the town is not easy.
Hungerford is, again, a very attractive town, but rather touristy and the High Street seems mainly to be given over to Antique and lifestyle shops. However, the nearest town with good shopping, Newbury, is easily accessible via the A4. And the shopping in Newbury is good, It is my preferred shopping town, although I live closer to Oxford.
Both are accessible by train. Whitchurch has trains to Basingstoke and Waterloo, Hungerford trains go to Newbury-Reading-Paddington and the West country.
Looking at the map, if you want to be equidistant between your children, Whitchurch and Hungerford are probably as good as anywhere.
You obviously have happy relations with your children, why not a park yourself on each for a week and try the journey and explore the towns you are interested in, see what is going on in these towns that interest you. Check houses out on RightMove. I travel regularly from Abingdon to Portsmouth and to York so regularly travel the route you would use and know the area quite well, so PM me if you want further information.