Although I'm afraid I don't have any experience with reaper, it is interesting that the issue you have described is pretty much the same as I experienced with Logic Pro X before I found a solution. Based on what I now know about Logic I'm wondering whether your issue has an equivalent cause, ie plugin(s) on existing tracks when you're recording a new one. With logic it seemed to be specifically a limiter plugin that had a lookahead element which caused the new recording to be offset from the existing tracks by a fraction of a second.