For example, on Another World, Bill Wolff announced all cast changes prior to the episode's prologue or at the beginning of the first scene in which the particular character was featured. When this happened(which was a lot, by the way, Wolff narrated these words:
Permanent Recasts: "The role of [character] is now being played by [actor]." Occurred when a actor was replaced on a permanent basis.
Temporary Replacements: (1) "Due to the illness of [regular actor], the role of [character] is being played by [actor]." If a actor was sick. (2) "Today the role of [character] is being played by [actor]." Did any other shows do this?

I would say, yes, it was common. I watched the old Doctors reruns on Retro TV, and they had a lot of recasts, and the announcer would say the same thing: "The part of Doreen Aldrich is being played by Pamela Lincoln." (You can watch this clip if you want on youtube starting around the 6min40sec mark.) https://youtu.be/xyGpe3UKuic?t=6m38s

https://youtu.be/xyGpe3UKuic?t=6m38s

I heard an AW ep form 1969, and Bill Wolff says "due to the illness of Charles Baxter, the part of Fred Douglas is being played today by Whitfield Connor."

“Ryan’s Hope” used to do it at the end of its opening theme. And they’d do it for the first few days after the new actor took over the role.

“General Hospital” rarely recast, relative to its peers, during its 1980s heyday. But I remember when Jack Wagner was having a contract dispute and was out for over a month. Kevin Bernhardt stepped in, and got an announcement (“The part of Frisco Jones is being played today by...”) at the beginning of his first scene of every episode he was in. And Les Tremayne got the announcement when he filled in for David Lewis (Edward Quartermaine) for a few weeks during the same era.