First of all, are you certain that "expecting" is the best word to use here? Expecting something means that you strongly believe it will happen soon, or are in a position of authority to demand that it will happen soon. In this sentence, the 2nd meaning seems to apply. Are you in a position to demand the next story soon? If not, try "looking forward to" instead of "expecting".
That out of the way, you can speak this way in a casual setting. It would be natural to hear someone casually say, or casually write:
Your story is great! Looking forward to your next one!
However, this would not be correct grammar in a more formal setting, or for more formal writing. The second sentence needs a subject and a verb, which in this case would be "I'm" (I am).
In the casual sentence above, both the subject and verb were dropped from the sentence because they could be understood. In that sentence, the "I'm" is implied, but not said out loud (or written down). Like this:
Your story is great! [I'm] looking forward to your next one!
However, in non-casual settings, you really do need to say or write the whole sentence for the English to be correct.
Your story is great! I'm looking forward to your next one!
This is correct in all settings.
Your second sentence is also correct in a casual setting, but it requires an additional implied word.
Your story is great, [and I'm] looking forward to your next one!
Making the implied words explicit makes the sentence valid in any setting.
Your story is great, and I'm looking forward to your next one!