Are "way better" and "way more" correct?

Apparently, "way" was a contraction of "away". It took on an adverbial meaning in the early 1900s:

In the 19th century, “‘way” was also used for “away” in the sense of a great distance: “way towards Tupper’s Lake” (1849); “He sat ‘way under the mantle” (1888); “way below cost” (1890); “mere specks, ‘way down the road” (1927), and so on. (Note that some writers used an apostrophe to show the “a” had been dropped from “away.”)

In addition, since the 19th century both “away” and “way” have been used as adverbs to add emphasis. These usages are heard chiefly in the US, the OED notes, though it includes some British citations.

Here are some “away” citations: “away up in Canada” (1818); “away down east” (1825); “away back in 1840″ (1882); “away up in price” (1903); “away behind” (1906); “I’m away wrong” (1910); and “away down in the list” (1858).

And here are some “way” citations: “way over yonder” (1850); “way down south” (1851); “way down East” (1854); “‘way down amongst the roots” (1866); and “sick of it way through” (1908).

Finally, we come to the usage you’re talking about, and here’s where “way” and “away” part company. This is the adverbial use of “way” to mean “much” or “far.”

Way can be used as an adverb, but it is very informal. According to this entry:

— adv

1. informal a. at a considerable distance or extent: way over yonder b. very far: they're way up the mountain
2. informal by far; considerably: way better
3. slang truly; genuinely: they have a way cool site

It is not incorrect to say that something is way better or that you have way more of something, but it is not formal. It has also been in adverbial use for long enough that 80 year olds should accept it.

I think the reason "way more" or "way better" might sound incorrect is that the word "way" is also commonly used as a noun, as in "which way should I go?". You could substitute "which direction should I go?" for "which way should I go?" but heretofore I have not heard anyone say (even in slang) that something is a "direction cool site." (I might bloody well have to eat my hat in the future but for now I haven't yet heard that instance of delightfully creative usage stretching enjoying the slang category of acceptability.) If someone is associating the noun form of the word "way" in an adverbial context (just as a matter of familiarity or habit and not by way of linguistic analysis), then the ear might reject the usage based on the fact that something seems "incorrect" when a noun is substituted for an adverb.

That is how I determined why the slang use of the word "way" as in "way more help" was unpleasant sounding to me when I first heard it. So while admittedly the ear can be deceiving and is no substitute for rigorous linguistic analysis, the ear as one of our 5 sense instruments does alert us to external stimuli that are different than what we are used to and therefore provides us with departure points for further investigation.

Outside of North America, Way seems to have almost entirely replaced far (more), much (more), many (more) as it was hardly ever heard in Britain no so many years ago so it must be an Americanism that has gained currency because it can be substituted for the other three words. But it sounds terrible and I would never use it.