yes it probably is an over-used word. I think it all depends.
If we are seriously trying our best to write a good lyric then the particular usage you elaborate on might be one of many things that we would try to avoid doing.
In your two examples 'I just can't let you go ' seems ok to me whereas 'I just love you so much' doesn't sound good, and this is something that I think is more important, for me at any rate.
Some words or phrases just (there it is) don't sound right in a song, in my view.
It so happens that my current song contains the line
'Was/Is forever just something to say.'
I think that's ok in the context because 'only', for example, or 'simply' wouldn't scan, and omitting the word wouldn(t have the same meaning.
Similarly "I'm just waiting for a friend" by the Stones.
Probably my favourite use of the word is in 'How can You Mend a Broken Heart' sung by Al Green where he sings 'I just wanna, I just wanna, I just wanna.....' about six times before the 'live again' bit. Not really part of the original lyric, just Al getting emotional.
Yes, it can be bad, but surely there are worse lyric writing sins.
Saying that, I don't like rules.