That is a product of that mandrel set being weak and able to bend.
The drilling not being centered on the blank does not matter, because everything ends up round in the end off of turning round from the mandrel (with a good mandrel).
I hate to say, you will continue to get cheap results from the cheap mandrel. You would be best off to scrape up the money to invest in some straight shank mandrels and a collet chuck. 1/2" and 3/4" for deer calls possibly from Webfoot, and a PSI collet chuck would be your cheapest good deal for it.