I don't ask in a non-begging fashion. Not only do I beg, I announce before and after the fact that I'm begging.

Honestly, I can't think of a way to ask her for a layout without sounding begging, though you can avoid the whining by just asking nicely.

Ahem: Hey, Jazzy. I know you make great layouts, and, even though you don't know me, I'd like to ask for one, please. I'm sorry for bothering you.

And then if she says no, say "thanks" and go ask somebody else. That's not whiny.