Do you mean the House? I think the Senate would still have 2 senators per state, and state-level electoral results are unlikely to change if an alienated minority from one state secedes, in favor of joining a more (politically) similar state.
For example if upstate NY joined Pennsylvania, then the D senator from PA would very likely lose. Or if (Republican-leaning) inland California joined Nevada, then Nevada would probably tip red. Or maybe (heavily blue) Jacksonville and Gainesville get fed up with DeSantis, and leave Florida for Georgia, cementing GA as a solid blue state.
While your question is still an interesting thought experiment, it is worth noting that OP was referring to counties moving between two existing states, not counties seceding a former a _new_ state, which would certainly have different ramifications.
And are counties only allowed to do it if they are contiguous with their desired state?