I can't find anything what's on this in 2023 - is EU anyhow working towards and agreement with US to allow not to give away private data? - As I see on Wikipedia, so far only UK has got this agreement with US.
As far as the EU is concerned, the main issue with the CLOUD act is whether it's possible to allow (personal) data transfers to a country where that's the law whithout violating the GDPR. The Commission really wants to do it but its efforts to create a legal framework for that have been repeatedly thwarted in court.
Where we are now is that the Commission published a draft adequacy decision to replace the International Safe Harbor Privacy Principles and is trying to get it adopted. It's not the law yet but there is no indication the Commission would be working on something else as it already has an agreement. Schrems and the EDPB seem unconvinced and it remains to be seen how this plays out when it ends up being challenged in some way.