Can you add a door to the bedroom hallway?

Are the bedroom doors solid wood or hollow core?

Adding more fabric will help absorb sound. For example, a thick hallway rug. But it won't make as much difference as adding doors, especially solid ones.

I grew up in a house like this and heard lots of interesting things I had no business hearing, after I'd been put to bed down the hall!

Plants do a pretty good job of absorbing noise. I like snake plants, they have a good color, aren't messy, and are really easy to care for. (They are toxic to pets and curious humans if eaten.)

Also, fabric wall hangings can absorb sound as well.

Or portieres (door curtains) instead of a doorway to the hall. They can be modern. Less permanent than a door, and probably less expensive.

https://fioritointeriordesign.blogspot.com/2015/05/the-elegance-of-portiere.html

You can purchase acoustic panels specifically designed to dampen noise. Some of these are strictly utilitarian, but some of them are kind of artsy.

Other than that, fabric and other soft things help. Art on canvas, a rug in the hallway, etc.

How high is the ceiling in the living/ dining space ? I agree more solid doors for bedrooms and a nice plush runner down the hall will help . It is always a bitof an issue when you have really high ceilings and then some low ones connected . I need to see your space to understand why what you have in the LR/DR is not absorbing more sound.I guess I wonder if you ever did the window covering

Books on shelves will absorb lots of sound

Wood in large cabinets & sideboards will absorb sound

Tree plants with lots of leaves help absorb sound

Drapery panels that have a lining will help absorb sound

Large upholstered seating including ottomans will absorb sound

If you have an area rug, make sure it's thick and place it on a felt rug pad to absorb sound

Large canvas art on walls absorbs sound.

Three dimensional art rather than prints under glass with frames help break up sound waves.

What is in the hallway? A runner, art on the walls? This might help. What art is on the walls in the living room? Thick padding under the rugs?

We hung portieres (door curtains) as @partim suggested in an open doorway to a small study and it made a huge difference