If you want people to write a letter of reference or nomination for you,

Give them lots of advance notice. It's rare that you hear of some award or job a week ahead. Usually it is many weeks or months. If you are urgent, draft a complete letter, so all they need do is to modify it. In fact, draft a letter or at least talking points in every case.

Make sure they know you well enough to write.

Make sure they owe you something. If not, they may decide that other tasks are more urgent. (That means that years before: you have been a fine student, an excellent assistant, whatever...)

It is not at all fatal that the letter discusses your weaknesses. Gives it credibility. The crucial part is your strengths. On the other hand, plagiarism is probably a fatal flaw, although suitable contrition may be worth noting.

Good letters are either detailed and specific and extensive in their considerations, OR they are much shorter but specific enough for the reader to appreciate the message.