It's not elegant, but it's valid - the "I" is effectively redundant, because "am" only ever refers to the first person singular. In some languages (such as Greek) it's a normal construction, because verbs are rarely ambiguous.
It's grammatically correct but, for what it's worth, it means you don't have a proper sentence as sentences need subjects for their verbs.
This sort of thing is normally email or "text speak" where no one uses proper sentences anyway so is appropriate for that informal style. At least the words that ARE there are spelled properly.
There is the lovely black country expression tha breaks the rule of am always following I 'You am a prat'.
I know it's not correct but I love that accent.
The omission of words (that would otherwise be required by the remaining elements) is called ellipsis. Elliptical construction is not uncommon in English and the OP's example of the missed personal pronoun would not be out of place when used for brevity in informal communications.
To quote The Beatles:
'Woke up, fell out of bed. Dragged a comb across my head.
Made my way downstairs and drank a cup.'
"Might be correct but it is lazy" is also an elliptical construction perhaps more properly (less lazily!) written as "It might be correct but it is lazy" ;-)
If I were tempted to convey this message and decided to omit the 'I', I would also omit the 'am'. If the circumstances are such that the recipient knows who is writing, there's no real need for either. "Having a good time" and "Just leaving work" are perfectly clear in text-speak...or shd I hve written txtspk?
We see this on postcards as you can get more written on it. You know the 'Having a lovely time, wish you were here' and on birthday cards, 'wishing you a lovely birthday' etc