tldr; There are two different processes created by the dual extrusion wizard and changing one doesn't always change the other. Rerunning the dual extrusion wizard with the template again and checking the settings to make sure they are as expected for both print processes fixed my problems.
I have been struggling with similar issues for several days now and the changes I've been making to my process to try and fix the stringing just haven't seemed to be making any material difference. I had a breakthrough on it this afternoon and wanted to share in case it helps out.
When I ran the dual extrusion wizard in the first place, I based it off one of my single extrusion profiles that has been working fairly well. When I had problems with oozing and stringing (specifically between the ooze shield and the main part), I started trying to adjust the settings in the process without rerunning the dual extrusion wizard. What I didn't realize was that the dual extrusion wizard had created two processes (one for each extruder) and I was only changing one of the processes and the other one ended up with a totally different set of settings.
I've played with it a bit since then and I'm not sure what the exact steps are to get the configuration mismatched between the two processes but I've managed to do it several times. Sometimes, it seems to apply the changes in one process automatically to the other but there's some set of steps that allows it to get totally out of sync. I think it has something to do with selecting a different profile in the drop down in one of the processes and then trying to change it back but I'm not totally sure.
In terms of a fix, the most reliable method I've found so far is to make the changes to the profile in the process, update it (or save as new), delete the processes and the process group, and then rerun the dual extrusion wizard with the revised profile set as the template.