In ordinary speech, there would not be a pause or gap between "living" and "out." They would form a continuous sequence of sound. How you choose to split this sequence into syllables is more a matter of phonological theory than of phonetics. There is no consensus on how English syllabifies consonants that come between two vowels.
You may have heard /n/ rather than the "ng" sound; this is not related to syllabification. It's an example of "g-dropping," where the suffix "ing" is pronounced with /n/ rather than the "ng" sound. This can occur regardless of the following sound--it would also be possible in "living room."