i think it is rude being angry at them for not offering.
I do not see asking as "begging" or "belittling"....no more would I see it as any other negative.
I don't have family near by...but when we go to stay with them (once a year), I always ask mum to babysit for the afternoon so DH and I can go to a movie or something. My mum is staying with us this week...and I will ask her to watch the kids for a few hours one weekend so DH and I can go somewhere together...i would never expect her to offer or get cut if she didn't offer.
I have had to ask friends to babysit DS before (during IVF and also when i had some pregnancy issues and DH needed to take me to hospital)...and never did i feel like I was begging or somehow demeaning myself. I take care of one of my friends son often during the school holidays (she works, i don't) and she asks me each term ... i do not see her as begging either.
I think it is perception...asking for help is not an awful thing...the strongest people i know ask and accept help all the time.