Learn English – I’d want to, not common or wrong

I would tell your students it's a "trap word," that is, something they might hear when conversing with native speakers but something that others might find jarring or unacceptable. (English has a handful of these – another that I can think of is "The data is..." vs. "The data are...")

As for Mr. Check New Dictionaries, I cringe when someone is dishing out grammar advice while using "u" for you and "n" for and. That undermines his credibility. Moreover, as for his assertion that it's "valid," he's partly right, and partly wrong.

When I consulted the WordNet 3.0 dictionary, for example, it said:

good ADVERB

(often used as a combining form) in a good or proper or satisfactory manner or to a high standard ('good' is a nonstandard dialectal variant for 'well');
- Example: "the children behaved well"
- Example: "a task well done"
- Example: "the party went well"
- Example: "he slept well"
- Example: "a well-argued thesis"
- Example: "a well-seasoned dish"
- Example: "a well-planned party"
- Example: "the baby can walk pretty good"

So the crux of the matter would be this: is ‘valid’ an accurate synonym for ‘nonstandard dialectal variant’? I don't think so. Like that fellow said, it's a matter of choice and habit – but a lot of habits are bad habits.

Both "I won't be needing that" and "I don't need that" are grammatical.

The problem with your teacher's logic is that "need" is not a modal in either of those. Two easy-to-detect signs of modality are:

• They have the syntactic properties associated with auxiliary verbs in English, principally that they can undergo subject–auxiliary inversion (in questions, for example) and can be negated by the appending of not after the verb.
• They do not inflect (in the modern language) except insofar as some of them come in present–past (present–preterite) pairs. They do not add the ending -(e)s in the third-person singular (the present-tense modals therefore follow the preterite-present paradigm).
  _Wikipedia

For the first point I listed, it should be pretty obvious that not does not follow need in "I don't need". For the second point, since people say "he needs...", we know it's not modal in this sense either.

"Need" specifically is a semi-modal. Your sentence needs modification were it to use need as a modal:

I need not have help.

According to the statistics here, the modal form of need is not used very often in American English (which is why I think it sounds old fashioned). A COCA search for BE needing (capitalizing "be" means it matches all forms of the verb, such as be, were, 're, etc.) returns 356 results, so I think it's safe to say that educated speakers do use "will not be needing" and similar (I would).