Often when asking someone for something point-blank, there is an obligation created for them to help you because you have no other person who you could feel comfortable asking. Since this is the case with you, no wonder you feel somewhat awkward putting someone in this situation!
Perhaps you need a way to ask that leaves the person an "out" if they do not want to help you - or cannot help you because they are not a suitable person to provide the help? If that is the case, you can ask in a way that is indirect. Then the actual asking is an aside to the help of being able to ask.
They can help you articulate what to say when you do ask the specific
person you know - exactly HOW and WHEN to ask them, and how to
tactfully make it OK for them to say no or to express that their
participation may be voluntary. Essentially, you're asking them for the
"inside information" of how to spring the question.
Examples of this approach might be:
"I am asking you because you might help me think of someone who could help me with ________. I'm having trouble feeling comfortable asking ______. Perhaps it would be best if the person would be in _______ position to help me. Would you know someone who is in this position? Let me tell you a little more what I need, so we can think of who might find it easy to help me."
Of course, while they're thinking about it with you, they then can volunteer to help you themselves.
Part of the "pay-off" of providing help to someone is the volition to want to help them. If you ask, (or demand) help or any gift in particular, and then specify what you need, it can be awkward and perhaps can cost the friendship if you pursue the question. (Specifying often needs to be done when asking for help.) This specific process of asking can take away from the recognition that the person has respect, rights, etc. Of course, respecting the capacity of someone to help you is the solution to this issue.
Asking indirectly is also a way to say how important to you this help would mean. You can express gratitude without being thought "one down." Losing status because you need help is a common assumption about the relationship between helper and "helpee." There is not a possibility of being a helper without the person to help, but our culture tends to regard those who need help as having something wrong with them.
One solution to this issue is to present the offer of help as a continuation of being able to give back to the person who you helped to continue the reciprocation of the relationship. "I enjoyed helping you, so I thought you might enjoy spending some time to help me. Might that be fun for you? Do you have time right now?"
I have more suggestions on this issue, but I don't want to make this too long...
If I have a problem with outright asking someone for help with something, I usually use a variant on the approach you suggested by describing the help needed and what I have thought of already as possible ways to approach getting the job done or help needed and then ask what the person I have approached might be willing to add. I have thought of this approach as "compatible or connected brain-storming." Your description of your particular approach is very helpful. Thanks!
P.S. Have you found that, in general, men prefer to give advice, not respond at all or indicate they are "blank" on the subject while women tend to give examples or comments on how we have or might handle similar situations or to offer concrete suggestions?