When writing dialogue, do I always have to follow a question with the verb "asked" or can I use "said" too?

This is an issue of the dialogue tag—the language after a line of dialogue that indicates who did the speaking:

"Will the flight leave soon?” he asked.

Here, “he asked” is the tag.

While you can certainly use “asked” as the verb in a tag that follows a question, you don’t have to. “Said” is a perfectly acceptable replacement. To say, after all, means to express oneself in words and we do that when we ask questions.

Keep in mind, not all lines of dialogue need to be followed by tags. You might use action to indicate the speaker and also reveal more about the circumstance or the character’s emotions:

“Will the flight leave soon?” He gripped the handle of his suitcase and looked out at the empty tarmac.

That action reveals more anxiety than this one:

“Will the flight leave soon?” He slumped in the chair and yawned.