Emma - I've thought a lot about your problem since reading it a couple of days ago, 'cos I'm tackling the same thing.
I'd get this overwhelming desire to stop and walk - in training runs and in races. So I tried to think it out logically. It happened when I was not particularly distressed by effort - usual training pace of 70% working heart rate, 3:3 breathing - no sweat. I wasn't ill - usual good health despite slight overweight and excess alcohol intake.
Therefore: it wasn't physical, it must be mental.
I looked at the "self-talk" when I wanted to stop - the "tapes" that were playing in my head....
If I was running with other people , I was saying "I won't be able to keep up, I can't keep up, I must walk..."
If I was running alone I was saying " So many more miles to go, I can't keep this up, I must walk.... or, I'm running well, it can't last, I'll get tired, I must walk..."
and lots of variations on these themes, so: I started a remedial program of running alone, starting with about 30 mins and working upwards. I purposely went slowly, just a tad above the effort needed to sit in an armchair. The sole objective was to move along continuously (apart from stopping to cross road junctions which is unavoidable round here). I play different "tapes" in my head about felling comfortable, feeling strong, and going steadily.
It's working. I'm now back to running 8 miles continuously at a steady pace, in comfort.
Everybody is different, but you may be able to take some elements of this and adapt them - sounds as if you're beginning to crack it already. Good luck.