BC - Below are screengrabs from the 1960-66 FAM and some web pics of an NOS control on ebay. It looks like you have to push the unit straight back into the dash cavity, probably might help to remove the glove box for an extra hand on the back side, then once clear of the dash opening angle it down and out under the dash.
The actuator cables may provide a lot of resistance to pushing the unit back, and defroster ducts might affect pushing it into the recess too.
Hope this helps.
Edit note: Don't forget to unplug the controls lamp/socket at the top rear of the unit. Maybe best done again by removing the glove box insert.
