If I haven't brought my textbook to class,
which of the following should I say:

"I forget to bring my textbook." or
"I forgot to bring my textbook."

The first one seems illogical because the problem
is "I forgot something when I left home" instead of
"I forget something at this moment".

Which sentence will native speakers usually use?
Your intuition serves you well.
The latter.

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Thanks a lot.
Does this reasoning apply to "lose" too?
i.e. I can't say "I lose my watch".
I should always say "I lost my watch."
Well, yes. It would apply to any verb, in fact. If it's in the past, it would be quite normal to use past tense.