Can I skip "to make" here? How is it change if I skip it?
Answer from a Native speaker
"To make" here expresses the verb action of delivering or announcing the announcement. It is okay to omit "to make" and just say "I've got an important announcement", however, there is a slight difference in nuance without "to make". When "to make" is included, this emphasizes the action and necessity of telling the announcement. Without "to make" here, it does not express the action of telling the announcement, and so it implies that it is not that necessary or does not need to be told immediately.