Is the young man considered sinful for falling under the hadith "None of you should propose to a woman to whom another has already proposed" in the described situation, given the girl's and her family's prior desire for her to marry her cousin and her refusal of that, and his proposal to her without a clear answer from her father?

What is known as a love relationship between young men and women is something not approved by Islamic law, and it is obligatory to cease conversing with the girl. If the girl refuses her paternal cousin's proposal, there is no harm in proposing to her, and the decision to accept or reject rests with her if she is not coerced. If she desires to marry you and you are a suitable match for her, her guardian does not have the right to prevent her, otherwise he would be an oppressor (عاضل), and she has the right to bring her case before a judge.