Real good or really good? Part-time or part-timely?

[VividJailbird]

My English teacher told us that, verbs should be followed by adverbs.

By I keep hearing Americans say:

It's real good.
It's real cool.

and:

I work part-time.
I work full-time.

Shouldn't they be:

It's really good/cool
I work part-timely/full-timely.

I'm so confused. And I wonder do Britishers speak like that too?

[Soup]

Standard American English:

Ex: really good, really cool
Ex: part-time, full-time

The adverbial suffix -ly is often dropped in speech.

To my knowledge, there part-timely, full-timely are not English.