Seems like it is worse as I work toward the tailstock.
Is it a Oneway or a Oneway clone as comes with some lathes such as Jet and Powermatic? Maybe a photo?
I agree with the suggestion to borrow a center and test to see if it behaves differently.
Are you turning spindles between centers or with one end mounted in a chuck? Diameters, lengths? Maybe double-check the center point alignment on the headstock/tailstock. I almost never use a cone center, but with any live center the alignment can be critical when holding one end in chuck and the mounting sequence might be important.
If it is a Oneway or clone, as someone mentioned, you can constrain the rotation with the supplied rod through the holes to remove the cone. Another use is to make special-use attachments that screw onto the threads on the ends - drill and tap a block of wood to fit (3/4x10tpi I think) then put a piece of soft iron wire through the holes so I can mount the live center in the headstock and turn the block as needed. (Hold with drawbar.)
Another thing - if the center point is knocked out from the back it reveals a #0MT socket - can turn a variety of attachments that way too - easier, cheaper, and more compact than the threaded block method. Turning a #0MT attachment is quick and simple.
This might be useful or at least interesting.
These live centers have two sealed bearings. I have no experience with removing/replacing a bearing in a live center. Google showed some videos and a statement that bearing replacement may not be simple.
Might be easy to find a replacement since some people have spares - I recently gave away two of the clones. (All those I've had are black, identical size and design, no markings.)
BTW, if you end up being in the market for a new live center I highly recommend the Nova - has so many advantages it's the center I use the most (if not using steb centers)
JKJ