I think we could solve that issue if we put our heads together.

For example We could admit new states as pairs to maintain the balance of power.

i don't think so, if it looks like the new state is going to trend blue then the red states are not going to admit it to the union. If it looks like it's going to go red then the blue states are not going to admit it either. If it goes perfectly "purple" or in between then neither side will vote to allow it because it dilutes funding. I don't see a new state joining the union ever.. same goes for one state splitting into multiple or multiple states consolidating into one.