Re: getting my three year old to walk and not insist to be carried
one of these: http://www.hippychick.com/index.cfm/children/Products.List/category_id/65
might be worth investing in to save your back a bit. (first site I came to when I googled).
My 3rd daughter wouldn't walk, she'd throw herself on the floor and scream blue murder, I only had her walking because her buggy had had it but ended up buying a new one at age 3 just for a quiet life. She did walk quite happily after a few months so maybe just back in the buggy for a while (long enough to forget about this phase) is something to think about.