Dice the onion into small, even pieces and measure out your minced garlic. Cut the cream cheese into small cubes—this is important because smaller pieces will melt much more evenly and quickly when added later, avoiding any lumpy texture. Add the chicken breasts, diced tomatoes with green chiles, cream of chicken soup, cream of mushroom soup, diced onion, minced garlic, and Italian seasoning to your slow cooker. Stir everything together to combine the ingredients evenly, ensuring the chicken is coated with the liquid mixture. Cover and set to cook on high for 3-4 hours or low for 6-8 hours, depending on your timeline.
Once the chicken is fork-tender (it should shred easily), remove the breasts from the slow cooker and shred them using two forks or a shredding device. Return the shredded chicken to the pot and stir it back into the sauce. I like to add the cream cheese cubes directly into the warm pot rather than mixing them in a bowl first—the residual heat melts them faster and more completely. Add the diced cream cheese and shredded cheddar cheese, then stir thoroughly to combine. Cover and cook on low for 20-30 minutes, stirring occasionally, until both cheeses are completely melted and the sauce is smooth and creamy.
While the cheese is melting in the slow cooker (or while you’re waiting for the chicken to finish cooking if you’re ahead of schedule), bring a large pot of salted water to a boil. Add the spaghetti and cook according to package directions until it reaches al dente texture—this is crucial because the pasta will continue to soften slightly when mixed with the warm sauce. Drain the pasta thoroughly in a colander, shaking off excess water. Add the cooked spaghetti to the creamy chicken mixture in the slow cooker and toss everything together until the pasta is evenly coated with the sauce. The dish is now ready to serve directly from the slow cooker, or transfer to serving bowls.