I went to grumble to my physio once about a similar thing. He said "you're in your late 30s, you're trying really hard at a physical sport, and you're complaining that you're trashed afterwards? Right. Have a hot bath and a rest"
If I was trashed in the same way as everyone else seems to be I wouldn’t be concerned
I always get this and did throughout my 30s. If I go for a hard day climbing my back and shoulders are always super sore the next day. Feel stiff and tired. Doesn't feel like DOMS to me which I was used to from weightlifting days.
I think it is to do with tightness/instability in some postural muscles leading to over-compensation from other muscles.
I've never solved the problem and tend to just live with it but I am one of the most injury prone people I know so not a good benchmark. The mobility exercises people are suggesting seem like a good shout.