It sounds like your tailstock is offset front-to-back some. That's not unusual, most tailstocks have that capability so you can turn long tapers.
Lots of "degrees of freedom" need to be considered when adjusting a tailstock. But, based on your latest comment, it appears you are pretty close to getting it sorted out.
Running a DTI along the quill will tell you if the tailstock has nod or rotation but it won't tell you anything about a height problem or if the tailstock is offset front to back. So you need to test with the centers AND a DTI.