I learned in a Gemora that the Kohanim stopped duchaning using the Four-Letter Name as spelled about 30 years after Herod rebuilt the Temple. I presume that when the Leviim sang Shir, they also used the Four-Letter Name, not pronounced as Adnus, but as spelled, and this also stopped. Is there any source for that as well?

I haven’t found an explicit source. Though it looks like it will apply to all mentions of Hashem's name. (see Moreh Nevuchim – the guide to the perplexed – chapter 62)