We did a wonderful 28 day drive tour of England and South Wales in October 2017.
Here is my detailed review of our trip.
28 days in Britain and Celebrity Eclipse home
Responding to your specific interests, I recommend you consider visiting Warwick Castle. Warwick is east of Stratford Upon Avon and north Oxford.
As for castles, consider including visiting Wales. Wales has an incredible number of castles. It goes back to the history of Wales when Wales was independent of England and England was trying to incorporate Wales into its control. Many castles were built in Wales, especially near the English-Welsh border.
Check my review and you will find some of the castles that we visited in South Wales.
There is so much to see in Britain that 10-12 day will only capture a small part of the wonderful countryside.
I highly recommend visiting York for old cities, it is wonderful. It still has its ancient walls that date back to the Romans.
The cathedral there is special (Minster). There is a fantastic National Railway Museum there. There is a lot more.
Bath has the Roman Bath Museum and more. Stonehenge is a special part of English history.
Regarding driving in the UK, I have some tips
If you aren't used to driving on the left, then rent a car with an automatic transmission. You want as few distractions as possible.
Also, suggest the vehicle include a navigation system, since it is very easy to get lost only using a map.
Try to find Bed and Breakfasts for lodging that are near the city or town center that have parking. Some places you will have much difficultly finding street parking.
Also, do not plan to drive too much in one day. Try to keep your daily drive to 2-3 hours, so you don't waste time on the road.
The British equivalent of our interstate highways are easy to handle (they are Dual Carriageways). The road is multilane and divided.
You will find many Roundabouts or traffic circles in Britain. You drive into them, yielding to those vehicles already in the circle. The circles go counter clockwise. Sometimes you have double roundabouts with traffic lights. They get more complicated, but just take it easy and you will be fine.
British drivers are very polite and not aggressive.
When driving on roads in the countryside you may run into narrow roads with hedges or stone walls that may not give you a lot of space if another car approaches. Just slow down and take it easy.
Carry coins for parking and the toilet.
Don't miss the Fish and Chips, the British Fish and Chips are amazing.
The Garden you mentioned is in the very north of England. If you go to York, past that city and past Hadrian's Wall.