Yes, you have the right information. As you said, Gyeongbokgung closes every Tuesday and Changdeokgung every Monday. Except the designated closing day, the palaces open everyday including national holidays. It means that there will be no other holidays for the palaces in October.

For your information, Changdeokgung allows visitors on a limited basis and Japanese guides are provided five times a day at 09:30, 10:30, 12:30, 14:30 and 16:30 from March to October. However, on Thursdays, you can move around the wider area of the palace and explore it more freely. To visit the palace on Thursday, you are advised to bring your passport since you will be required to leave your identification document at the ticket office if you borrow the free Japanese audio guide.

As for Gyeongbokgung, please make sure to arrive at the palace before 5p.m. Even though the palace closes at 5:30 p.m. visitors are not allowed to enter the palace after 5p.m.

Thank you and welcome to Korea.