Correct, they may have legitimately come from Saxony in northwestern Germany near the border with holland. The Angles came from extreme northern Germany on the Denmark border; it is certainly possible if not very probable.
The Saxons that came to England were actually a confederation of previous tribes. Much like the Franks who were really a conglomeration of previous tribes.
The original Saxones were on the German-Danish border near Schleswig. The Chaucii were more settled in NW Germany in modern day Lower Saxony, closer to Holland.
When warfare became more and more prevalent among the Germanic tribes, old tribal divisions broke down and new ones sprung up that were usually more military oriented.
The saxones and chaucii , as well as some Bructeri became the Saxons or Saxon confederation.
Tribes like the Cheruscii, Marsii, Angrivarii and Usipatii became the original Franks.
Tribes further south, like the Marcomanni, Mattiaci, Hermandurii, and Chatti, became Allemanns
when you hear about the Battle of the Teutoberg Forest in 9 A.D., there is no mention of Saxons or Franks, but of Chaucii ,Bructerii, Cheruscii etc..., who were in the same location as the later Saxons and Franks.
It was before the old tribal divisions had broken down. The Chaucii, Bructerii, and Cherusci would later make up a large part of the Saxons and Franks.
