Is there a way to cover carpeting so it gets less dirty?

It was very popular when I was growing up, still available. Plastic clear runners or larger. Check with your local flooring or big box stores.

I too was going to suggest the clear runners. Place them over the high traffic areas.

Runners in high traffic areas helps. Consider replacing them in the future with wood or laminate. I only have area rugs now and my life and cleaning is 100x better and easier.

Clear runners or carpet runners will save your carpet from dirt, but can mat down fibers over time. Prohibiting footwear and food/drink in carpeted areas is much easier.

What are you using to "wash" the carpet? Incorrect cleaning products will actually attract more dirt and grime. I have mine professionally steamed cleaned every 3 months, but I have a house full of pets. An average family should have done once a year to maintain and get optimum length of wear. Or, purchase a steam cleaner for a couple hundred dollars and do it yourself.