Looking for dog owner advice with Bermuda grass

Looking for any input from anyone who has lived in North Carolina long enough to know. I recently moved here in 2025 and this is my first winter. My dog absolutely loves rolling around in the grass outside. This was never an issue when I used to have fescue as the grass never really stuck on his fur. Now with the dead winter grass, my dog gets absolutely covered in the Bermuda grass and brings it inside. I’m wondering if it’s because I never cleared my grass clippings over the summer? Or it just seems like Bermuda is super brittle and just breaks off into tiny pieces no matter what during this part of the year.

Is there something I should be doing differently when I mow my lawn and or / prepare for winter?

Is the grass just something dog owners deal with?

Not complaining or anything. Just trying to educate myself on my new location.

Thank you!