I recommend getting a larger DR rug. If someone is sitting down, their chair would be half on, half off the rug.

Can you open the blinds and take better picture of the LR? Are there any accent colors or art you could pull inspiration from?

Yeah, we need more light to see your colors. Also, what kind of thing do you have in mind, and what do you want to spend? I have found that Wayfair and Houzz are both places that are good for viewing and buying rugs. There is a universe of style options out there. Do you like traditional oriental designs? Graphical moderns? Stripes, shapes, florals, solid colors, washable, wool, indoor-outdoor, loop, pile...? See what I mean?

I woudn’t bother with a dining area rug as the space is too small to use the appropriate size.

As for you living room rug, go to a showroom and ask to see “solution died nylon”. It won’t shed and should sweeties have an accident the solution dye fabric will stand up to cleaning better.

a lot of the time new freshly cut rugs, especially wool rugs shed for a while. constant vacuuming really helps. it stops shedding eventually and if its nice it might be worth to stick out the shedding process.