I am not sure that I can use only present simple in this case.
I should have taken my umbrella before I go/went - present simple or past
The keys must have fallen out of my pocket before I get/got in the the the car
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The verb tense of the subject should be consistent with the verb tense of the dependent clause.
Thus, the correct sentence from your first example is:
I should have taken my umbrella before I went.
'should have taken' - past tense
'before I go' - present tense
'before I went' - past tense
Think of it this way: If you 'should have done' something in the past, you cannot simultaneously 'should have done' it in the present. You cannot be in two places at the same time.