Q:

Is this quote correct?  Can one live in enjoyment or be heartful? 

I just came across this quote;

Your heart is the only one that can show you how to live in
  fulfillment and enjoyment.  It's time to become heartful. 

Is it even possible to 'live in enjoyment'?  Or to 'become heartful'? 
These were the definitions I found of both the noun and adjective of heartful:
Noun: As much as the heart can hold or contain; as much as a person wants or can endure. 
Objective: Characterized by deep emotion or sincerity of expression; genuine, sincere, heartfelt.  Of an emotion: deeply or acutely felt; intense. 
Are there any more definitions?  It seems to me that it does not fit in the sentence at all this way. 

A:

You're right about both of the things you hesitate to accept, at least in standard English goes.  However, we should probably understand that heartful is gaining a new meaning. 

How to live in fulfillment and enjoyment. 

Enjoyment requires a complement.  The Oxford English Dictionary licences this observation:

enjoyment, n.
   1.  The action or state of deriving gratification from an object.  Also, in weaker sense, the possession and use of something which affords pleasure or advantage.  Const.  of. 

That means you can't just say enjoyment any more than you can use enjoy as an intransitive verb.  You have to know what is being enjoyed.  Here's a good example from the OED:

1849 [..] He would protect the Established Church in the enjoyment of her legal rights. 

The word we would normally use in the sentence you quoted is joy. 

It's time to become heartful. 

Although heartful does appear in dictionaries, it's not very common.  This probably accounts for the author's slight misuse of the word.  It applies to emotions, expressions, actions — not people. 
Here's the OED again (which I note you quoted in your question):

heartful, adj. 
  Characterized by deep emotion or sincerity of expression; genuine, sincere, heartfelt.  Of an emotion: deeply or acutely felt; intense. 

Here are some of the OED's more illuminating examples:

1881 [..] The heartful prayers, the fireside blaze and bliss. 
1983 [..] Joseph and Nathan sing in a heartful union of two voices with one loving vision. 
1997 [..] Jones combines the literal with the lyric, street-smart eloquence with heartful passion. 

It's hard to know exactly which sense is intended by the author.  But I might suggest passionate, sincere, compassionate. 
That said, as you observed, this word may well be undergoing a shift.  It has that parallel with mindful that you noted, which suggests the technical use specific to a particular community or subculture. ( Moreover, in your cited text it clearly echoes heart at the beginning.) 
Languages are always changing and we have to avoid judgements about a term like "even though it doesn't really exist" once we see it in wide use. :)