OK, so cox.help@cox.com confirmed for me that I do not need a modem change but they could not help with the actual changing of the plan. They gave me the number to the Cox Loyalty Team, 1-888-438-6673, and they made the change without any resistance at all. They were strangely accommodating actually. I went from $87.99 per month for the Internet Preferred 250 plan to $70.49 per month for the same Internet Preferred 250 plan. I did find an email from Cox stating, "Faster internet is now on us." So as it turns out I was on a 150/10 plan originally at $69.99/mo. After a year they bumped the price to $87.99/mo. Then the sneaky devils converted it into a 250/250 plan and sold it to me with a "look how cool we are, we just upped your speed at no extra cost!" But like slithering snakes in the grass they quietly offered it at $70.99 to potential new subscribers (yes, my new plan is discounted a whole $0.50 from the advertised price). I bet there is market data that shows a large swath of existing customers never notice decreased costs in subscriptions they're already in and thus continue to pay the same inflated price.
I appreciate you participating in my "struggle" - lol. Hopefully someone else finds this thread helpful.