What is the ruling on designating the new mosque for Friday and Eid prayers, and the old mosque for the five daily prayers, knowing that the new mosque was built adjacent to the old one to accommodate the increasing number of worshippers?

What you are doing is correct. If the old mosque is spacious enough for the worshippers during the five daily prayers, then it is better to limit yourselves to praying there, to increase the reward for praying in it, and for the reward to continue for its builder. Not everything should be expanded and demolished in the construction of mosques. There is a right for the predecessor who built the mosque first; he has more right to it. It is not permissible for anyone to demolish his construction or to spoil his reward.