Late one night over Memorial day weekend I finally had a chance to take the movement apart and it was as dirty as I suspected that it would be; after all, this movement was commonly used by soldiers during the second war (is that the great war? 'are't they all?'). The first problem that can be seen is something binding in the winding and setting assembly and it turned out to be Rodicao, likely from the last person working on it to prevent the clutch spring and other pieces from being lost.
I move on to taking everything else apart to be cleaned as it is a rather dirty movement and in dire need of a cleaning. As I was taking it all apart I discovered that the staff was straight and that the hairspring was ever so slightly bent out of true which I believe has been causing the balance to oscillate at an unpredictable rate. I unfortunately I do not have truing calipers. In the spirit of learning and impatience, eyeballing it it is with tools on hand! Onward with what is likely a bad idea.
I came across an article http://mb.nawcc.org that talks about how it is possible to bend the hairspring dog legged around the out of true portion and it should still oscillate at the correct frequency as the overall dimension/weight does not change. With the article I read as hope of a fix, I plow forward and remove the hairspring from the balance and after hours of trial and error and constant worry of making it worse, I manage to bend the hairspring while keeping it on my steel block which has a trued surface to aid in keeping the shape of it as true to it's self as possible.
The watch runs a bit better but it will now need either a new mainspring or a hairspring, or both as it will run for a few hours than stop after many times of experiencing pucker factor 5 while trying to true the spring without the proper tools. It was an interesting experience to use every tool I have at my disposal and even modifying some to work better for this specific use in the name of Horology. Hurray for a mostly working Doxa for now and onto hunting for a mainspring that will fit it.