According to The UVic Writer's Guide, "Anyway" means "anyhow" or "in any case." "Anyways" is a strictly colloquial expression, as ungrammatical in written English as "anyhows" because adverbs cannot be plural.
And The Columbia Guide to Standard American English says that Anyways is dialectal, primarily Southern and South Midland. It raises Standard users’ eyebrows nearly everywhere else in all speech and is unacceptable in Edited English. Use anyway instead, which American English spells as one word (I think I’ll stay home anyway) and stresses like other compounds. The two-word phrase (adjective modifying noun) is not a compound: You can use this gadget [in] any way you like.
Also read the explanation from "The Mavens' Word of the Day" below:
anyways
Ben Wise wrote: "As ugly and juvenile as it sounds to these ears, which had almost never heard the construction until, perhaps, a decade or two ago, anyways now seems to have taken hold as common usage among the current generation of youth and young adults, at least where I live (New England). My guess is that it is a "juvenilism," like, you know, retained into later years by today's subliterate culture. I'm curious to know whether it was ever a generally accepted regional form anywhere, or anywheres, and if my cringing impression--that it is effectively and universally replacing anyway--is correct. Any ways you might devise to answer this would be appreciated."
It's not just youth who are using anyways; a quick search of the news from the last month turns up citations of everyone from a 7-year-old girl to Prime Minister Jean Chretien of Canada. What is clear, though, is that nearly every citation is either from direct speech that is being quoted, or from Web sites that are not edited. Anyways is still not accepted in edited prose--and my word processor's spell checker will also not recognize it.
You may be hearing anyways more often simply because of modern media, especially television. When print was the primary medium of mass communication, dialectal differences tended to be edited out. Television and radio now broadcast unedited speech from people from all over the world, and so perhaps anyways is being picked up from there. However, it is being used exclusively in its logical position--as a conjunctive adverb--and not as a substitute for the determiner-plus-noun construction of your last sentence.
Anyways, often spelled any ways, is a dialectal variant of any wise, 'in any way/manner', and as such is recorded from the 16th century, in such august tomes as the 1611 King James Bible and the 1560 Book of Common Prayer: "All those who are any ways afflicted...in mind, body, or spirit." As an adverb, its formation from any way by the addition of the genitive -s is perfectly regular: we got always from alway in the same manner. This use of anyways, along with the use of any wise/anywise, is now obsolete.
However, the use of this genitive form instead of the more usual conjunctive adverb anyway still survived in certain dialectal uses, among them New England dialects. Opinions about the use vary; the fact that Noah Webster recorded anywise as "sometimes used adverbially" in 1828, but did not record anyways, may suggest that he disapproved of anyways, although it's also possible that he never heard it used or chose not to focus on the obscure dialectal variant. The 1914 edition of The Century Dictionary records both meanings of anyways, and calls them "colloquial in both senses." The first edition of the Oxford English Dictionary calls the conjunctive use "dialectal or illiterate" (the same harsh judgment you levied).
So we're left with the fact that the use has actually been around for quite a while (at least the late 19th century, and likely to be much earlier), and that it has never been considered standard. However, as a dialectal variant, it is not incorrect; it is simply a less frequent use. I have my own theory, completely unsubstantiated, that there's a Cockney connection lurking in there somewhere. If you've ever heard anyone with that accent, you'll know that anyways is pronounced like anywise. There just has to be a link between the British settlers of New England and the regional pronunciations of their mother country.