You can't use the seals from one grand company to pay for something that belongs to another one. You keep all GC/Allied Seals as well as your rank with that particular Grand Company, but you will be unable to access something from the new Grand Company until you've risen your rank with that specific faction to a satisfactory level. This includes the quests and hunting log requirements for promotion.

As a side note, don't change grand companies while you're doing Atma Books because some of the Leves are Grand Company leves that require you to be a Chief Sergeant with your current faction in order to undertake.