Everything, every verb must be in present tense when writing in present tense?
That could not be more incorrect. There are twelve different expressions of tense in the English language, 4 each of past, present, and future (some linguists say there are fourteen), and all of them can be expressed in the same scene, and more than one can be expressed in the same paragraph, and even two (rarely) can be expressed in the same sentence or clause. And this applies if your manuscript is written in simple past tense or simple present tense or in a hybrid of past and present.
Assume your story is written in simple past tense, and includes this line of dialogue: "Mom says we are allowed to watch TV until eight," said David.
Well, guess what—that contains two verb tenses: 'are' (present) and 'said' (past) in the same 12-word sentence.
Even in a past tense story, dialogue is spoken from the temporal position of the speaker, which is always from their present. This proves that not all verbs must be past tense in a simple past tense story. So the short answer is yes, you absolutely can use both past and present tense in a past tense story.
What about a story told in present tense? It depends. If you are using the ubiquitous present tense format rampant in a YA or middle grade novel, the tendency is to use nothing other than present tense verbs. But this does not mean you have to do that. If that seems to be a rule to you, the truth is you can break that rule any time you wish, legitimately, as long as you do it consistently. Because that rule is false.
It honestly does not matter how adamant your creative writing teacher is about following the YA format. If Charles Dickens can write a book like Bleak House (which will be in the canon forever) in present tense and NOT use the YA conventions rampant today, then you can, too.
Thinking that to tell a story from the perspective of the protagonist being live in the scene means that all you need to do is use only present-tense verbs in place of where one would use past-tense verbs is not an effective way to tell a story, regardless what you are being told. Even John Updike couldn't make that work believably.
But there are ways to use both present and past-tense verbs together to do exactly that—express a story or scene as if it is happening right now, rather than being a telling of a story from the past. It has the advantage of no distance compared to past tense stories, and it has the advantage of not being fuzzy and ridiculously confusing and amateurish-sounding that the YA present tense convention has.
It can also be invisible instead of sounding like an egoic author affectation or a gimmick.
I have five novels written this way that are living proof of this.
Some think the rule is 'don't mix tenses'. But that is NOT a rule. The rule is, much more accurately, 'Use the proper tenses in the proper places'. Since each verb modifies a subject or object apart from what every other verb in a sentence does, you can use both, properly, together, without 'mixing' them or having them clash.
The reason your line ending in 'he asked me' works, is because a line of dialogue is an incidental event and not an ongoing condition. Once the line has been delivered, it is over, meaning it is not precisely in the present. It's now in the immediate past, and can and should take a past-tense dialogue tag. The dialogue tag is there to inform us who said the line that was just delivered, and not there to inform us who the line is being said by directly at the moment it is said.
And if you do it this way, it still maintains the 'live in scene' aspect, completely.
It is logistically impossible for events even in a present-tense story to be expressed only by present-tense verbs, because even then there is a moving timeline where incidental events are immediately pushed into the immediate past. Everything that is observed by a protagonist is reported to the reader after it happens, not as it happens, which would also be logistically impossible.