Ohhh! They’re Trailer Trash!!. Why didn’t you say so. 
By the looks of the OP’s second post I think Illegitimate was absolutley the right word to use 
The problem is that despite it’s dictionary definition, as Cellphone points out, it definitely has a negative connotation to it in ‘modern’ usage. Illegitimate has the whole “three kids by three different men, trailer trash” vibe to it. (Which while apparently correct in this instance could be viewed as offensive).
As to ditzy, no way is that offensive, but then again as has already been said it seems to be a term reserved exclusively for women, so who am I to comment? but it seems to me to be a very tame word, and while it is not really a positive word, it lacks that slightily nasty undertone of bimbo.
I would be more predisposed to take offense to “trailer trash” than the other words. Neither the mobile-home park where the grandmother Maureen lives now, nor the one whose rent she paid for her daughter over a period of several years, is or was “trashy”–in fact Maureen’s place cost her over $100,000. As a frequent visitor to both I can attest to the pleasant ambience they have. I myself have lived in a mobile-home park for 20 years and I don’t consider ours–Garden West Estates in Gardena–“trashy,” either.
I am positive at this point that “criminal” is a more fitting term to describe Adelaide and Natalie, political correctness notwithstanding.