Are you detected whilst cloaked more quickly if you run?
Some areas in the game are labelled as 'Distrust Area' under the compass. Using the cloak allows you to blend in and slow the detection.
Do Saxon soldiers detect you more quickly if you run instead of walk?
Yes is the answer. When walking past guards my 'attention meter' was small and white but as soon as I started running it turned yellow and filled very quickly.
The guards soon attacked when the bar filled and my disguise was lost.
So when you're in these areas wearing your cloak, just walk and keep your distance from guards.