I love home-made egg noodles with my sooup. I always add the noodles after the rest of the soup is cooked and seasoned the way I want it. It comes out great.
My biggest mistake with homemade woup came when I was asked to bring a big pot of my chicken noole soup for a fundraiser to help a family. I made the soup to the best of my ability. I also made the home-made egg noodles and added them to the soup. I cooked them until they were soft and then put the whole thing in the refrigerator as I had to take the soup to the location during my lunch hour. Like yours, my noodles had become mush. The flavor was good, but the noodles were indistinguishable from the rest of the soup. When I stirred the soup, the noodles became a thickening agent. I learned that lesson the hard way. I now add noodles just before serving, as I do with the meat.
I cook the meat (cubed turkey, beef, chicken, etc.) seperately from the soup, as boiling meat will toughen and dry it out. I flavor the broth with bones and skin (for poultry), or bones and fat (for beef, and pork). Then I get it cold and lift the hardened fat from the gelled stock. I strain it and heat it to a low boil. Add the veggies and flavorings and cook until done. Then add the noodles, dumplings, or whatever pasta needs to be incorporated. And finally, when serving, add the cently cooked meat cubes. That way, everything is at the proper texture, degree of doneness, and at its best flavor.
Hope this helps.
Seeeeeeya; Goodweed of the North