Sometimes this kind of thing is hard to find. That can be due to a too big influence of one of the groups or a too low... or also because the vertex (or vertices) are not in the group.
To check that, you can go to edit mode, select one of the vertices involved in the problem and have a look at the panel on the right :
This panel allows you to change the values so that you can understand the way the weights need to be changed.
Note that you can select several vertices, but only the weights of the active one are given. But in this case, the panel allows to copy the weights of the active vertex to the selected ones.
About normalization : in my experience, yes this can be important to keep it. If you don't you may have unwanted results.
But normalization won't remove 'all assignation'. Normalization is a calculation so that the sum of the vertex weights is 1.
When you paint, you may have weights values like bone1 = 1 and bone2 = 0.6.
Normalization will do so bone1 = 1 / (1 + 0.6) and bone2 = 0.6 / (1 + 0.6) so that bone1 + bone2 = 1.