very minor spoiler but.... They were building both at different locations. Explained in one of the books i think.
I don't think this is accurate. The Death Star II was much bigger than the first, and also did not have the exhaust port weakness. It was an improvement over the design of the first.0
i swear one of the books mentions that they were being built a different locations. Something about the Emperor always having a backup plan. Maybe it was the just the superlaser they were talking about. It has been a while. That Superlaser shows up about 4 times in the subsequent books.
They must have ordered an Acme Death Star from the sears calendar or something. Either that or all the Mexicans that were thrown out of Arizona were flown to space for construction.