What rule does the sentence "Before you were born, I am an adult," violate?

Jesus’ famous claim in John 8:58 has been translated into English as “Before Abraham was, I am”. Without going into too much detail - it harkens back to Israelite history when God revealed himself to Moses with “I Am” as a name or title. Jesus’ use of the term was enough for the temple authorities to charge him with claiming to be God. The construction of the sentence was therefore probably as surprising in Greek as it is now in English.

Unless you are making a similar claim, the point of dissonance in your own reply is that you have mismatched temporal (timing) cues.

The first part of your sentence sets up the frame of reference in the past: “before you were born”. The second part speaks about the actual present “I am an adult”.

You can normally pick any point in time as your ‘now’ moment and align the rest of your tenses to that time. By setting up your sentence to implicitly reference the actual present time as your literary ‘now’, your tenses all reference the present.

The first part of your quote “Since ...” speaks about a period beginning in the distant past and continuing into the present. Your claim in the second half should match this period with something like “I have been ...”. Instead, it ignores the past and makes a claim only about the present: “I am ...”. This sets up the dissonance you picked up.

The above preserves the first half of your sentence at the expense of the second half. It is possible to wrench a preservation of the second half at the expense of the first.

You can interpret the starting “Since” as “Because” instead of as an indication of duration, though you’d need to ignore the temporal reference “before”. This would be an unusual interpretation in the given context, but it would at least make sense of the present tense claim. Interpreting it that way claims that you are an adult because the other person was born. This is an awkward interpretation, made worse with “before” included in the sentence.

All told, the sentence sounds odd because of its mismatched temporal cues.