"Wait, is it douzo? In the book it says dōzo..."
It's both. And it's neither!
In the beginning stages I use rōmaji (English letters) to write Japanese in class. This is to give you a head start in learning to speak.
Some people think you shouldn't use rōmaji at all, because it will give you bad pronunciation.
That might be true if you're studying by yourself.
But if you have a teacher to teach you how to pronounce Japanese words correctly, and correct your mistakes, you can learn Japanese correctly using rōmaji.
I'm pretty strict on pronunciation, I think. My students have good pronunciation - even the beginners.
Anyway, there are different systems for writing Japanese in the English alphabet. Depending on which system is being used, a word could be spelled quite differently.
どうぞ (do-u-zo) means "here you are" / "go ahead".
Some writing systems use a macron (a horizontal bar over the letter) to write the long vowel sound: ā ī ū ē ō. Then, it would be written "dōzo".
Another method is to spell out the letters: aa, ii, uu, ei, ou. That gives us "douzo".
Both "dōzo" and "douzo" are correct.
Sometimes, the long vowel isn't written in: "dozo". This is wrong!
You might also see ee and oo used instead of ei and ou: "doozo".
Personally I think "doozo" is just asking for trouble. That's not how the word is spelled in Japanese (it's どうぞ do-u-zo).
Of course, the only truly correct way to spell the word is to write it in Japanese: どうぞ.
But both "dōzo" and "douzo" are fine too. Just don't forget the long vowel!