In this link, I found the word "sit tightly",but when I look for it in dicitionary, I get the idiom "sit tight". Are there differences between "sit tight" and "sit tightly"?
Sit tightly and sit tight mean two entirely different things and you cannot use one in place of the other. The book you are reading uses sit tightly correctly.
We are sitting at ease; if a fourth person comes, we will be sitting a little more closely, more tight and it will be really good. It is getting cold, and to sit tightly, closely, will be warmer.
To sit tightly here means for several persons to sit close together. One cannot sit tightly on their own. It is synonym of "sit closely together". The room is small and the night is very cold, if they all sit closely together, they'll be warmer.
To sit tight means "to wait patiently, not to move". As in the example given in the Cambridge dictionary:
You'd better sit tight and I'll call the doctor.