I have a CX bike (or an adventure bike as they are often marketed nowadays) and it's fantastic. Love love love it. A pure CX race bike would probably be a lot less comfy over any distance than my bike, also proper CXs tend not to have the lugs for a bottle cage as adventure bikes do.
I have taken my roadie up the Marriotts Way several times, with no ill effects. Along the Bure Valley way too, also with no problems. They are well trodden cycle paths/footpaths/bridle paths. Having said that, there are bits of M.W. that I wouldn't take a roadie along. I wouldn't want to take a roadie around some of our Broads pathways though.
Test riding is going to be the key here, and asking your bike shop a LOT of questions. Have you any thing in mind at the moment?