What you are going through is not normal behavior.
The 5600x doesn't keep boosting until it hits 95C the way the ryzen 7000 does.
In an ambient of 32C, idle shouldn't be more than 50C. Either your cooler isn't making adequate contact or its a voltage issue.
I would say open the instruction manual again and make sure its the correct mounti6ng hardware, spacers and the correct backplate.
My cpu block came with its own backplate and little plastic washers. I thought the backplates are the same and used the stock mobo backplate. Ended up with issues very similar to yours.
The threads tightened down but the mounting pressure wasn't enough. I could barely tell the two backplates apart visually. With the correct backplate installed, temps didn't go as high and even took longer to hit max. Wasn't like an instant jump to 95C. That usually means a bad mount.
Check for a bios update as well. If its a voltage issue, that will help.