I am using QGIS 3.26.1. I am having difficulty with label callouts. I have used this feature many times over the past few years. At this time when I adjust a label with a callout, only the callout line is moved, the label is treated as a separate object and does not move as a unit as it has in the past. The label has to be moved separately so that it is at the end of the callout.
I have this issue too, where the sensitivity of the callout line is higher than the label itself. When I try to move the label (and hope that the callout line automatically follows), I often end up moving the callout line manually instead.
This is especially the case with short labels where the label anchor point is Centroid. After the label is initially moved to a manual location, it becomes nearly impossible to select the label instead of the leader line. See below for an example:
The easiest workaround is to switch off visibility of callout lines in the label settings before you're ready to move any labels, then move the labels as required, then switch the callout lines back on again, so you don't accidentally move the callout lines instead of the labels.
Another option is to temporarily change the label anchor point setting to Closest Point or one of the other options, which may make it easier for your mouse cursor to select the label. However, I dislike this approach as it changes all of the leader lines' anchor points, and sometimes that's visually distracting and I lose track of exactly where I meant to move my label (easier to switch it all off instead, at least for me).