Do people in China say 新年快乐 at Spring Festival?
Yes. 新年=New Year, while it does not specify whether it's the new year of the Chinese calendar(农历) or the Gregorian calendar(公历).

So it's ok to say 新年快乐 in both cases. And normally we'll interpret it based on the context. If you wish to be more specific, you can use 元旦 for 公历新年 and 春节 for 农历新年.

Edit:

My personal opinion about the usage of 快乐 in greetings (as a native speaker myself):


Generally, 新年快乐 seems more formal to me while 新年好 seems more casual.
新年快乐 is mostly used with other greetings/blessings, starting with 祝(你). eg. 祝你新年快乐，万事如意，心想事成. Thus, it's better used in written language, where you'd use one long sentence to express your best wishes to someone.
Both can be used when people meet during the Spring Festival, but you'd use multiple short phrases instead of one long sentence in the oral language, so 新年好 probably better suit the case.