I'm trying to use the adjustment brush to overexpose the background of a photo. I select the adjustment brush, paint the background, and then adjust the exposure, but the exposure for the entire photo is changed.
Quite often people are caught out by a quirk of the interface: the brush adjustment panel can be expanded or collapsed.
When it's expanded, you can see its whole separate set of sliders. When collapsed, these sliders are not visible and the Basic panel's sliders are easy to mistake for those. Changing any basic panel slider has global effect even if you are also in the middle of working on a local adjustment.
The triangular icon button to expand and collapse the local adjustment panel, is found to the right of "Effect". The latter is where, even when the panel's collapsed, a local adjustment preset can still be chosen.