Often, someone will say:
I'm not living in a senior's home!
When the intended meaning is:
I will not be living in a senior's home!
Is this acceptable?
Your two quotes don't actually have the same meaning. "I'm not living there!" is close to "I refuse to live there", while "I'm not going to be living there" is a prediction. Technically, the first is "I will not live there", while the second is "I shall not live there", but I doubt whether many people these days appreciate the difference, particularly with exclamation marks and apostrophes involved.