How do I 'de-formalize' my dining room set to fit into my new home?

I find that reupholstering something or painting the upholstery can completely change a piece of furniture. See this tutorial for an example of painted upholstery fabric: https://www.11magnolialane.com/tutorial-how-to-paint-upholstery-fabric-and-completely-transform-a-chair/)

You could also paint the wood on your chairs/table with chalk paint or something like that. That would give it a nice, rustic look.

I have painted our dining room furniture to give it a different look, I never thought my tastes would chaneg over the years but they definitly have!! https://vickymyerscreations.co.uk/thrifty/restyled-shabby-chic-dining-room-chairs/

There are lots of posts on Hometalk which give step by step instructions for painting chairs https://www.hometalk.com/search/posts?filter=paint%20a%20chair

Do any of these match the style you have compared to the style you want? https://www.hometalk.com/search/posts?filter=dining%20room%20furniture%20makeover

I just read a good blog post about how to make your new furniture match your old pieces, I think that the same concept would apply to matching furniture to your new space. She painted hers, which is a pretty easy way to update pieces to match your new style https://farmfreshvintagefinds.com/how-to-paint-your-existing-furniture-to-match-your-new-furniture/