At a certain point, if Musk consistently blocks things favoring one party and not another does X risk losing its Common Carrier status and start opening itself up to more liability?
If you're referring to US law, Twitter was never a telecommunications common carrier. So it can't lose that status through any sort of moderation decisions.
Twitter (I still refuse to call it X) doesn't have "Common Carrier" status, no website or platform does. Section 230 of the Communications Decency Act of 1996 protects platforms from liability for user content even if they display bias towards that content.