Sorry for delay in getting back.
Your DD is on the north side of the river and there are several villages nearby: Mordiford (good pub and lovely wood nearby. ), Fownhope, Lugwardine, Bartestree. None are what you would call rural. There is a health centre with swimming pool etc within easy reach of all of these villages.
I am more familiar with the villages south of the river.
Little Dewchurch is a lovely little village with a good community feel. Kingsthorne, Much Birch and Little Birch are all nice villages which segue one into the other. All have lovely views but are only 10-15 minutes from town (Hereford). There is an excellent doctor's surgery in Much Birch. Much Dewchurch is also a good place to be.
Heading further south, there are the villages of Wormelow (has a shop - open 7.30 to 9pm, as also is the Post Office there) , St Weonards (also has a shop). Really rural are the villages of Orcop and Garway, which is an extraordinary place with a pub, small primary school and amazing new community centre built with lottery money - it is a "passsiv" building and is an architectural gem - loads of stuff going on there all the time, and a great community feel. I am always driving over there!
Good luck with your searches. Lots of the villages have their own websites, so it is worth looking at these.